NATIONAL  DRAWING  COURSE. 


TEXT-BOOKS. 
Free-Hand   Drawing. 
Mechanical  Drawing. 
Color  Study. 
Light  and  Shade. 
Historic  Ornament  and  Design. 


(In  preparation.) 
(In  preparation.) 


TEACHERS'   MANUALS. 

Outline  of  Drawing  Lessons  for  Primary  Grades. 
Outline  of  Drawing  Lessons  for  Grammar  Grades. 

DRAWING  CARDS. 
National  Drawing  Cards  for  Primary  Grades. 

DRAWING  BOOKS. 

One  book  each  for  the  4th,  5th,  6th,  7th,  and  8th 
years  of  school. 

SPECIAL    MATERIAL    FOR    THE    NATIONAL 

DRAWING    COURSE. 
The  Cross  Transparent  Drawing  Slate. 
The  Cross  Pencil  for  use  with  the  slate. 
The  National  Drawing  Models. 
The  National  Model  Support  or  Desk  Easel. 


MECHANICAL   DRAWING 


A  MANUAL  FOR 


BY 

ANSON   K.  CROSS 

Instructor  in  the  Massachusetts  Normal  Art  School,  and  in  the  School  of  Drawing  and 

Painting,  Museum  of  Fine  Arts,  Boston.     Author  of "  Free-Hand  Drawing, 

Light  and  Shade,  and  Free-Hand  Perspective"  and  a  Series  of 

Text  and  Drawing  Books  for  the  Public  Schools. 


BOSTON,    U.S.A. 
GINN    &    COMPANY,    PUBLISHERS 


COPYRIGHT,  1895 
BY  ANSON   K.  CROSS 


ALL   RIGHTS    RESERVED 


PREFACE. 


THE  following  notes  are  intended  for  students,  and  for 
teachers  of  elementary  work,  particularly  for  public  school 
teachers.  There  are  many  books  on  the  subjects  of  projec- 
tion and  working  drawings,  but  none  which  present  the  prin- 
ciples in  ways  suited  to  the  needs  of  the  large  number  of 
teachers  who  are  required  to  give  instruction  in  these  subjects 
in  the  public  schools.  Most  of  these  teachers  have  had  little 
instruction  in  the  subjects  and  frequently  do  not  understand 
problems  which  they  are  expected  to  explain.  This  occurs 
because  too  difficult  work  is  often  planned  for  their  grades,  and 
also  because  the  instruction  which  many  public  school  teachers 
have  received  has  been  so  advanced  and  theoretical  that  the 
simple  principles  which  alone  are  necessary  for  elementary 
work  have  been  lost  in  the  attempt  to  understand  descriptive 
geometry  and  the  drawings  of  machine  and  other  details,  whose 
nature  and  use  are  often  not  known. 

To  understand  descriptive  geometry  certain  qualities  of  mind 
are  absolutely  necessary,  and  many  find  it  impossible  to  com- 
prehend even  the  simpler  problems  of  this  subject.  The 
draughtsman  or  drawing  teacher  of  advanced  work  who  is 
without  practical  knowledge  of  the  subject  of  descriptive 
geometry  is  very  poorly  equipped  for  his  duties ;  but  this 
knowledge  is  not  necessary  for  the  public  school  teacher,  who 
will  find  it  best  to  treat  the  subject  of  working  drawings  in  a 
much  simpler  way.  , 

This  book  presents  principles  and  not  a  graded  course  of 
lessons.  It  rovers  more  than  many  teachers  may  require, 

2065952 


IV  PREFACE. 

though  special  students  or  classes  of  the  high  school  may  study 
work  as  advanced  as  any  of  that  presented.  Teachers  of  draw- 
ing in  the  high  school  should  understand  all  the  problems  of 
the  book. 

It  is  hoped  that  the  book  may  assist  teachers  of  both  gram- 
mar and  high  schools  so  to  understand  the  subject  that  they 
may  give  to  classes  instruction  suited  to  their  capacity  and 
needs. 

The  author  desires  to  express  his  obligations  to  G.  A.  HILL, 
A.M.,  who  has  read  the  proof-sheets  and  furnished  him  with 
many  valuable  suggestions. 

ANSON  K.  CROSS. 


CONTENTS. 


PAGE 

CHAPTER  I.  —  MATERIALS  AND  THEIR  USES i 

PAPER i 

DRAWING  BOARDS 2 

PENCILS 3 

T-SQUARE  .  . 4 

TRIANGLES ,  4 

COMPASSES 5 

DIVIDERS 7 

NEEDLE 7 

SCALES  8 

FRENCH  CURVES 9 

PENCILING 10 

CHAPTER   II.  —  GEOMETRICAL  PROBLEMS 12 

CHAPTER  III.  —  WORKING  DRAWINGS 28 

NATURE  AND  USE  OF  WORKING  DRAWINGS 28 

STUDY  OF  PRINCIPLES 32 

MAKING  WORKING  DRAWINGS 35 

VIEWS  OF  A  CIRCULAR  PLINTH 36 

VIEWS  OF  AN  HEXAGONAL  PLINTH 37 

VIEWS  OK  A  PLINTH  AND  Disc 39 

VIEWS  OF  A  BOX  AND  PYRAMID 40 

DIMENSIONING *  •  •  43 

LETTERING  .  ' 45 

CHAPTER  IV.  —  DEVELOPMENTS 47 

THE  CUBE 47 

PRISMS 49 

THE  CYLINDER 49 

THE  CONE 50 

PYRAMIDS ...  51 

THE  SPHERE .  .  51 


VI  CONTENTS. 

CHAPTER    V.  — SHADOW  LINES 

CHAPTER  VI.  — INKING 

INDIA  INK 

INKING  A  DRAWING 

BLUE  PRINTS 

ERASING  AND  CLEANING 

SHARPENING  THE  PEN 

STRETCHING  PAPER          

CHAPTER    VII.  —  MACHINE  SKETCHING  AND  DRAWING 
CHAPTER  VIII.  —  ORTHOGRAPHIC  PROJECTION     .        .        .        .        . 

PROJECTION  PRINCIPLES 

AXES  OF  PROJECTION 

VIEWS  OF  A  POINT 

POSITION  OF  A  POINT 

VIEWS  OF  A  STRAIGHT  LINE      .        .        .        .        . 

VIEWS  OF  A  PLANE  SURFACE         .  

VIEWS  OF  A  SOLID      . 

VIEWS  OF  A  RECTANGULAR  CARD 

VIEWS  OF  A  PYRAMID         . 

TRUE  LENGTH  AND  POSITION  OF  A  STRAIGHT  LINE   . 

STATEMENTS  OF  PRINCIPLES 

PROJECTION  PROBLEMS    .'•.-• 

CHAPTER  IX.  — SECTIONS          .        .        . 89 

SECTIONS  OF  THE  SPHERE ...        90 

SECTIONS  OF  THE  CUBE     " 90 

SECTIONS  OF  THE  CYLINDER .        .        92 

SECTIONS  OF  THE  PYRAMID 93 

SECTIONS  OF  THE  CONE  .........        94 

CHAPTER  X.  —  INTERSECTIONS •.    -.     -    .        .        .96 

INTERSECTIONS  OF  A  LINE  AND  A  PLANE  SURFACE   ...        9*7 

INTERSECTIONS  OF  A  LINE  AND  A  CURVED  SURFACE       .        .        -99 

INTERSECTIONS  OF  SOLIDS  ....'....  102 
CHAPTER  XI.  —  ARRANGEMENT  AND  NAMES  OF  VIEWS  .  .  .  104 
CHAPTER  XII.  —  PLATES  AND  EXPLANATIONS  .  .  .  .  109 

DEFINITIONS ...  184 


MECHANICAL    DRAWING. 

CHAPTER   I. 
MATERIALS    AND    THEIR    USES. 

1.  GOOD  work  cannot  be  done  without  good  instruments.     The 
best  work  cannot  be  done  without  steel   T-squares  and   triangles, 
steel-edged  drawing  boards,  and  drawing  instruments  of  the  best 
make.     Students  of  art  and  technical  schools  should  provide  them- 
selves with  the  best  instruments. 

In  the  public  schools,  no  more  can  be  done  than  to  give  a  little 
knowledge  of  the  principles  of  instrumental  drawing,  which  will  be 
valuable  to  all.  To  do  the  best  work  technically  is  impossible, 
because  the  pupils  are  too  young,  because  they  do  not  have  the  time 
necessary  for  practice,  and  because  they  do  not  have  the  materials 
necessary  to  produce  the  best  work.  Any  one  of  these  reasons  is 
sufficient  to  prevent  the  making  of  first-class  drawings,  and  together 
they  make  it  impossible  for  us  to  expect  that  perfect  results  can  be 
obtained. 

Although  pupils  of  the  public  schools  labor  under  the  above-men- 
tioned disadvantages,  it  will  not  do  for  the  teacher  to  expect  bad 
work,  or  to  be  satisfied  with  work  which  at  first  glance  is  seen  to  be 
inaccurate ;  for  even  with  the  imperfect  materials  provided  it  is  pos- 
sible, by  careful  work,  to  obtain  drawings  which  are  neat  and  suffi- 
ciently accurate  for  the  requirements  of  simple  working  drawings. 

2.  Paper. —  The  paper  must  be  tough  and  should  have  a  surface 
which  is  not  easily  changed  or  roughened  by  erasing  lines  drawn 
upon  it.     This  is  most  important  when  drawings  are  to  be  inked. 
For  free-hand  sketching  a  soft  paper  is  best ;  but  for  all  mechanical 
work,  the  paper  should  be  hard  and  strong. 

For  pencil  drawings  a  paper  which  is  not  smoothly  calendered 
is  best,  because  the  pencil  marks  more  readily  upon  an  unpolished 
paper,  and  because  its  surface  will  not  show  erasures  as  quickly  as 


2  MECHANICAL   DRAWING. 

that  of  a  smooth  paper.  For  public  school  use,  several  kinds  of 
cheap  paper,  which  are  good  enough  for  the  work,  may  be  obtained 
both  in  sheets,  in  block  form,  and  also  made  up  in  blank  books. 

Whatman's  paper  is  the  best  for  drawings  which  are  to  be  inked. 
There  are  two  grades,  hot  and  cold  pressed,  suitable  for  this  use ; 
the  cold-pressed  having  the  rougher  surface.  If  the  paper  is  not 
to  be  stretched,  the  cold-pressed  is  preferable,  as  its  surface  shows 
erasures  less  than  that  of  the  hot-pressed.  The  side  from  which  the 
water-marked  name  is  read  is  the  right  side,  but  there  is  little  differ- 
ence between  the  two  sides  of  hot  and  cold  pressed  papers.  Stretch- 
ing the  paper  is  unnecessary  except  when  colors  are  to  be  applied  by 
the  brush,  or  when  the  most  perfect  inked  drawing  is  desired. 

The  use,  in  the  public  schools  below  the  high,  of  sheets  of  blank 
paper  instead  of  blank  or  other  drawing  books  has  its  advantages 
and  its  disadvantages.  If  a  drawing  upon  a  sheet  of  paper  is  spoiled, 
it  may  be  thrown  away  and  another  begun  upon  a  new  sheet ;  but 
this  fact  tends  to  careless  work.  To  the  teacher  loose  sheets  of 
paper  are  a  source  of  great  care,  even  if  kept  in  portfolios  or  large 
envelopes  and  retained  by  the  pupils,  for  the  drawings  must  be 
examined,  and  it  is  not  easy  to  keep  them  arranged  in  the  order  in 
which  they  are  made. 

Blank  books  cost  little  more  than  paper ;  and  their  use  tends  to 
neatness  and  care  on  the  part  of  the  pupils,  each  of  whom  is  inter- 
ested to  produce  the  best  book.  Drawings  in  the  books  are  always 
arranged  in  order  and  ready  for  examination.  The  chief  objection 
to  their  use  is  that  they  cannot  be  handled  as  a  block  for  free-hand 
purposes,  or  be  used  with  a  T-square  for  instrumental  work.  This 
difficulty  is  avoided  by  fastening  the  book,  by  means  of  two  rubber 
bands,  to  a  drawing  board  made  to  receive  it.  When  fastened  to 
this  board  the  book  may  be  used  for  instrumental  drawing  as  advan- 
tageously as  paper  upon  a  board.  Used  in  this  way  a  book  is  prefer- 
able to  sheets  of  paper ;  therefore  a  board  should  be  provided 
whenever  books  are  used,  whether  the  books  are  blank  books  or 
those  of  any  system. 

3.  Drawing  Boards.  —  Drawing  boards  should  be  made  of  clear 
white  pine,  and  should  not  be  painted  or  varnished ;  they  should 
have  cleats  upon  the  back,  s'o  that  upon  the  whole  working  surface 


MATERIALS  AND    THEIR    USES. 


of  the  board  the  grain  of  the  wood  runs  in  one  direction ;  for 
when  paper  is  stretched  upon  cleats  which  are  fastened  at  the  ends 
of  the  board,  it  is  often  spoiled  by  the  swelling  of  the  board,  which 
moves  back  and  forth  upon  the  cleats. 

The  cleats  should  not  be  glued  or  otherwise  firmly  secured  to  the 
board,  since  the  board  must  change  its  width  with  the  weather,  and 
if  the  cleats  are  firmly  secured  to  it,  the  board  will  split  or  warp. 
The  cleats  should  be  fastened  by  means  of  screws  and  washers,  the 
screws  being  placed  in  slots,  which  allow  the  board  to  move.  This 
construction  is  not  necessary  for  small  boards  required  in  the  gram- 
mar schools. 

The  best  board  for  use  with  a  drawing  book  is  one  provided  with 
a  groove  to  receive  the  head  of  the  T-square,  which  should  be  thick 
enough  to  remain  in  the  groove  when  the  book  is  between  the  square 
and  the  board. 

4.    Pencils.  —  To   do   good   instrumental   work   two   grades    of 
pencils  should  be  used,  a  hard  one  for  the  fine  working  lines  and  a 
softer  one  for  the  result  lines.     The  hard  pencil 
should  be  sharpened  to  the  wedge-shaped  point 
illustrated,  which  is  as  thin  as  possible  one  way, 
and  about  one-half  the  width  of  the  lead  the  other. 

The  softer  pencil  should  have  a  point  of  the 
same  shape  but  thicker,  so  as  to  give  the  width 
required  for  the  result  line. 

To  sharpen  the  pencil  a  knife  should  be  used 
to  give  the  wood  the  flattened  form  required  in 
the  lead.  The  lead  should  then  be  worked  down 
by  the  use  of  a  fine  file.  A  substitute  for  the 
file,  which  may  be  used  in  the  public  schools, 
is  given  by  gluing  a  straight-edged  piece  of  sand- 
paper to  a  strip  of  wood.  A  point  with  which  good  work  can  be 
done  cannot  be  obtained  by  the  use  of  the  knife  alone. 

A  pencil  with  a  round  point  should  be  used  for  all  free-hand 
lettering  and  figuring,  and  for  other  work  of  a  free-hand  nature,  such 
as  the  drawing  of  irregular  curves. 

The  hard  pencil  should  be  used  very  lightly,  as  much  pressure 
will  indent  the  paper  so  that  its  marks  cannot  be  removed, 


4  MECHANICAL   DRAWING. 

All  lines  should  be  drawn  with  the  pencil  slightly  inclined  in  the 
direction  in  which  it  is  moved.  The  pencil  should  not  be  moved  in 
the  opposite  direction,  as  it  will  then  act  as  a  plough  to  tear  the  sur- 
face of  the  paper. 

Any  and  all  lines  not  needed  in  the  finished  drawing  should  be 
erased  at  one  time  after  the  final  lines  have  been  determined,  for 
the  surface  of  the  paper  is  soiled  very  quickly  when  worked  upon, 
after  erasures  have  been  made.  The  working  lines  and  other  lines 
that  are  to  be  removed,  should  be  erased  when  the  drawing  is  ready 
to  finish  and  before  its  outlines  have  been  strengthened,  in  order 
that  the  final  lines  may  be  left  in  perfect  condition,  and  may  not 
require  retouching  on  account  of  use  of  the  eraser. 

5.  T-Square.  —  A  T-square  is  an  instrument  used  in  connection 
with  a  drawing  board,  for  drawing  straight  lines ;  it  consists  of  two 
parts,  a  blade  and  a  head,  which  are  secured  at  right  angles  to  each 
other.     The  blade  of  the  T-square  should  be  placed  upon  the  head, 
which  should  never  be  cut  to  receive  it,  and  should  be  secured  to  it 
by  means  of  screws.     This  construction  allows  the  parts  to  be  sepa- 
rated for  straightening,  and  the  triangles  to  be  moved  across  the  head, 
which  is  often  desirable,  but  cannot  be  done  when  the  blade  is  set 
into  the  head. 

The  T-square  should  be  used  for  drawing  horizontal  lines  only. 
Its  head  should  always  be  placed  upon  the  left  edge  of  the  board. 
Vertical  lines  should  be  drawn  by  the  use  of  a  triangle  placed  upon 
the  T-square  and  not  by  means  of  the  T-square  only ;  because  the 
edges  of  the  board  are  seldom  exactly  at  right  angles  to  each  other, 
and  the  blade  of  the  T-square  is  often  not  at  right  angles  to  the 
head,  so  that  lines  at  right  angles  to  each  other  will  not  result  from 
the  use  of  the  T-square  upon  all  the  edges  of  the  board.  Only  the 
upper  edge  of  the  T-square  should  be  used,  as  the  edges  are  often 
not  quite  straight  or  parallel. 

6.  Triangles.  —  The  usual  forms  are  illustrated.     The  45°  tri- 
angle has  two  angles  of    45°  and  one  of  90°.     The  30°  and  60° 
triangle  has  an  angle  of  30°,  one  of  60°,  and  one  of  90°.     By  placing 
these  triangles  upon  the  T-square,  lines  at  any  of  these  angles  with 
a  vertical  or  horizontal  line  may  be  drawn. 


MATERIALS  AND    THEIR    USES. 


Lines  parallel  to  any  given  line,  AB,  may  be  drawn  by  first  plac- 
ing together  two  triangles  (C  and  D),  so  that  a  side  of  one  (C)  coin- 
cides with  AB,  and  then  sliding  triangle  C  upon  triangle  D,  being 
careful  not  to  allow  D  to  move. 

Any  parallel  lines  are  most  conveniently  drawn  by  sliding  one 
triangle  upon  the  other,  or  upon  the  T-square. 


Lines  perpendicular  to  AB  may  be  drawn  by  placing  either  tri- 
angle upon  the  other,  so  that  its  hypotenuse  coincides  with  AB, 
and  then  revolving  the  triangle  through  an  angle  of  90°,  into  the 
position  illustrated  by  the  dot  and  dash  lines. 

Lines  at  30°,  45°,  and  60°  with  AB,  may  be  drawn  by  placing  one 
triangle  upon  the  other,  so  that  an  edge  coincides  with  AB,  and  then 
reversing  the  triangle,  as  will  be  shown  by  experiments. 

When  paper  is  to  be  cut  by  the  use  of  a  knife  and  straight  edge, 
the  T-square  or  triangle  should  not  serve  as  the  straight  edge,  for 
its  edges  would  soon  be  nicked  and  spoiled  by  the  knife. 

7.  Compasses.  —  Compasses  suitable  for  any  use  should  have 
jointed  legs,  which  will  allow  the  points  to  be  placed  at  right  angles 
to  the  paper,  whatever  the  size  of  the  circle  to  be  drawn.  Compasses 
should  not  be  used  for  circles  which  are  too  large  to  allow  the  points 
to  be  thus  placed.  A  lengthening  bar  is  generally  provided,  which 
greatly  increases  the  diameters  of  circles  which  may  be  drawn. 

The  joint  at  the  head  of  the  compasses  is  the  most  important 
feature.  It  should  hold  the  legs  firmly  in  any  position,  so  that  in 
going  over  a  circle  several  times,  only  one  line  will  result.  It  should 
allow  the  legs  to  move  smoothly  and  evenly,  and  should  be  capable 


6  MECHANICAL   DRAWING. 

of  adjustment,  as  the  parts  soon  wear,  and  the  joint  then  becomes 
too  loose.  The  joint  should  never  be  so  tight  that  much  pressure  is 
required  to  move  the  legs. 

One  leg  of  compasses  is  usually  provided  with  a  socket  to  which 
are  fitted  three  points  ;  a  divider  point,  a  pencil  point,  and  a  point 
carrying  a  special  pen  for  the  inking  of  circles.  Each  of  these 
points  is  generally  provided  with  a  joint,  so  that  it  may  be  placed  at 
right  angles  to  the  paper. 

The  other  leg  should  be  jointed ;  it  is  often  provided  with  a 
socket  which  receives  two  points,  one  a  divider  point,  and  the 
other  carrying  a  needle  point.  Such  an  instrument  may  be  used  as 
dividers  for  spacing,  or  as  compasses  for  penciling  or  inking  circles, 
and  will  be  all  that  is  needed  for  public  school  work.  The  pen  point 
is  not,  necessary  for  grammar  school  use,  as  inking  should  not  be 
attempted  before  the  high  school. 

The  cheaper  grades  of  compasses  are  provided  with  sharp,  pointed, 
solid  legs,  and  are  very  objectionable,  because  these  points  work  into 
the  paper  as  the  compasses  revolve  and  make  large  holes  which  spoil 
the  drawing.  Even  for  public  school  work,  compasses  should  have 
a  needle  point  with  a  shoulder  which  prevents  the  making  of  holes 
in  the  paper.  It  may  not  be  possible  to  provide  for  the 
grammar  schools  compasses  with  jointed  legs  ;  but  the 
needle  point  with  the  shoulder  must  be  insisted  upon,  if 
even  fair  work  is  desired. 

The  compasses  should  be  held  very  lightly  between  the 
thumb  and  forefinger,  and  should  be  inclined  slightly  in  the 
direction  in  which  the  line  is  drawn.  No  more  pressure  should 
be  applied  than  is  necessary  to  obtain  the  line.  In  inking,  little  more 
than  the  weight  of  the  compasses  is  needed  ;  but  in  penciling  more 
pressure  will  be  required  for  result  lines.  Two  grades  of  pencils  are 
more  necessary  for  use  in  the  compasses  than  for  straight  line  work ; 
they  should  be  sharpened  as  shown  on  page  3,  and  so  that  the  wide 
side  of  the  point  is  at  right  angles  to  a  line  extending  from  the 
pencil  point  to  the  needle  point.  As  this  is  difficult  to  do,  pupils 
below  the  high  school  may  use  a  sharp  conical  point, 


MATERIALS  AND    THEIR    USES.  7 

/The  legs  of  the  compasses  should  be  moved,  to  give  any  desired 
radius,  by  taking  hold  of  each  leg.  The  needle  point  may  be  placed 
on  one  point  of  the  drawing  while  this  is  done  ;  but  if  the  strain  due 
to  changing  the  radius  is  brought  upon  the  needle  point,  it  will  tear 
the  paper  and  spoil  the  drawing. 

8.  Dividers.  —  The   compasses   are   changed    into   dividers  as 
already  explained.     To  set  off  equal  spaces  on  any  line,  hold  the 
dividers  lightly  between  the  thumb   and  forefinger  and   place  one 
divider  point  at  any  desired  point  in  the  line  to  be  spaced ;  then 
revolve  the  dividers  about  this  point  as  centre  until  the  other  divider 
point  comes  to  the  line,  when  the  dividers  are  to  be  revolved  about 
it  as  about  the  first  point.     Revolve  the  divider  points  first  on  one 
side  and  then  on  the  other  of  the  line  to  be  spaced,  and  never  lift 
both  points  from  the  paper  at  the  same  time.     Continue  thus  until 
the  line  is  spaced. 

To  divide  a  line  of  given  length  into  any  number  of  equal  parts, 
by  use  of  the  dividers,  it  will  be  necessary  to  use  so  little  pressure 
that  visible  punctures  are  not  made  until  the  correct  space  has  been 
obtained ;  if  this  is  not  done  in  two  or  three  trials  a  second  line 
should  be  drawn  and  the  spacing  continued  upon  it.  If  one  line  is 
gone  over  many  times,  the  divider  points  will  fall  into  the  punctures 
previously  made,  and  accurate  work  will  not  be  obtained.  When 
the  correct  space  is  found  the  points  must  be  marked  in  the  line  so 
that  they  can  be  readily  seen.  To  do  this  the  line  should  be  gone 
over  several  times,  each  time  more  pressure  being  applied  to  the 
dividers,  until  the  punctures  become  visible.  If  the  pressure  re- 
quired to  place  the  points  is  applied  at  once,  equal  spaces  will  not 
be  given,  as  the  dividers  will  spring  and  move  while  marking  the 
points. 

To  do  work  of  this  nature  easily,  a  pair  of  spring  dividers  should 
be  used.  This  instrument  has  one  point  attached  to  a  spring,  which 
is  regulated  by  a  screw,  so  that  very  slight  changes  in  the  space  may 
be  made  with  ease. 

9.  Needle.  — The  needle  may  be  used  by  the  draughtsman  and 
by  advanced  pupils  with  great  advantage  as  regards  both  accuracy 
and  speed.     It  should  not,  however,  be  used  by  young  pupils,  or 
below  the  high  school. 


8  MECHANICAL   DRAWING. 

The  needle  may  be  used  to  set  off  distances  from  the  scale  and 
to  mark  the  intersections  of  lines ;  it  may  also  be  used  when  lines 
are  to  be  drawn  through  two  points,  to  hold  the  triangle  so  that  only 
one  point  requires  the  attention  of  the  draughtsman.  This  is  done 
by  placing  the  needle  in  one  point,  holding  the  triangle  against  it, 
and  revolving  the  triangle  until  it  comes  to  the  second  point. 

Points  should  never  be  marked  by  holes  in  or  through  the  paper, 
but  by  the  smallest  punctures  which  can  be  seen.  These  are  much 
more  definite  than  pencil  marks,  and  have  the  advantage  of  locating 
points  so  that  they  are  not  lost  by  erasures.  When  a  point  is  marked 
in  this  way  a  small  pencil  circle  should  be  drawn  about  it  free-hand, 
in  order  that  its  position  may  be  readily  seen. 

The  needle  point  may  be  placed  in  a  handle  of  soft  wood,  and 
should  project  just  far  enough  to  be  used,  but  no  farther,  as  acci- 
dents will  happen  if  it  is  not  carefully  handled.  The  best  place  for 
the  needle  point  is  at  the  unsharpened  end  of  a  pencil.  A  double- 
ended  pencil  holder  for  round  leads  may  have  the  pencil  in  one  end 
and  the  needle  in  the  other.  This  holder  should  be  about  4^-"  long, 
so  that  it  may  be  reversed  readily. 

IO.  Scales. —  When  objects  are  small  they  may  be  represented 
full  size ;  but  when  large,  the  drawings  must  be  smaller.  Common 
scales  for  mechanical  drawings  are  £,  |>  |,  and  -fa  full  size.  These 
scales  are  often  written  6"— i  ft.;  3*'=i  ft.;  i£"—  i  ft.  and  f"=i  ft. 

Large  objects  are  often  drawn  to  very  small  scales ;  in  maps  an 
inch  often  represents  many  miles. 

Instead  of  selecting  one  of  the  scales  named  or  one  found  upon 
the  ordinary  scales  used  by  draughtsmen,  drawings  may  be  made  to 
any  scale  whatever.  Thus,  if  any  object  is  to  be  represented  in  a 
certain  space,  a  scale  should  be  constructed  which  will  cause  the 
drawing  to  fill  the  space  in  the  best  way. 

To  determine  the  scale  by  which  the  drawing  of  any  object  may 
be  made  of  any  desired  size,  divide  the  length  of  the  object  by  the 
length  of  the  drawing  desired.  Thus,  suppose  an  object  21"  long  is 
to  be  represented  in  a  space  which  will  allow  the  drawing  to  be  io£" 
long.  The  drawing  must  be  half  size,  and  may  be  made  by  measur- 
ing the  lines  of  the  object  and  making  those  of  the  drawing  half 
as  long.  If  the  drawing  can  be  but  7"  long,  it  will  be  one-third 


MATERIALS  AND    THEIR    USES.  g 

size.  To  make  the  scale  for  this  drawing,  draw  a  line  4"  long  and 
divide  it  into  twelve  equal  parts,  which  represent  inches.  Divide 
one  of  these  spaces  into  eight  equal  parts  to  represent  eighths  of 
inches.  By  means  of  this  scale,  the  drawing  may  be  readily  made 
by  taking  from  it  the  dimensions  of  the  different  parts. 

If  views  are  to  be  made  of  a  mallet,  the  length  of  the  front 
view  to  be  6",  while  the  actual  length  of  the  object  is  15",  a  scale 
from  which  the  sizes  of  the  different  parts  can  be  taken  may  be 
made  by  drawing  a  line  six  inches  long  and  dividing  it  into  fifteen 
equal  parts,  each  part  representing  one  inch.  One  space  may  be 
divided  into  eight  equal  parts,  and  by  means  of  this  any  part  of  an 
inch  may  be  obtained. 

The  triangular  scale  (architects')  has  upon  it  the  following  scales  : 
-5TZ1  ix  T3sj  i>  i»  i>  f  >  x»  *i  and  3  inches.  These  are  not  provided  and 
are  not  necessary  in  the  grammar  schools,  where  a  cheap  foot  rule 
having  three  or  four  scales  ranging  from  £  to  f  full  size  will  generally 
be  all  that  is  required.  If  a  drawing  cannot  be  made  full  or  half 
size,  or  by  one  of  these  scales,  a  scale  can  be  constructed  by  the 
pupils,  as  explained. 

When  views  of  objects  are  to  be  made,  they  should  be  of  such 
size  and  so  arranged  as  to  produce  a  pleasing  effect  upon  the  sheet 
or  the  page,  and  when  none  of  the  scales  provided  will  give  the  best 
size  to  the  drawing,  a  special  scale  should  be  constructed. 

ii.  Irregular  or  French  Curves.  —  These  curves  are  not  re- 
quired below  the  advanced  work  of  the  high  school.  They  are  used 
for  curves  which  cannot  be  drawn  with  the  compasses. 
To  draw  an  ellipse  or  other  curved  line  by  their  use, 
care  must  be  taken  to  have  the  French  curve  exactly 
cover  as  many  points  in  the  line  as  possible,  and  then 
only  the  central  points  should  be  connected.  If  the 
line  is  drawn  to  all  the  points  covered  by  the  French 
curve,  it  will  generally  fail  to  continue  smoothly 
through  the  points  in  the  line  beyond  the  curve. 

When  French  curves  are  to  be  used  the  lines 
should  be  very  lightly  sketched  free-hand  through  the 
determined  points,  before  the  curve  is  applied  to  the 
drawing  to  obtain  the  final  lines. 


!O  MECHANICAL   DRAWING. 

12.  Penciling.  —  Penciling  is  generally  done  to  prepare  for  ink- 
ing or  for  the  making  of  tracings  on  tracing  cloth,  but  sometimes 
practical  shop  drawings  are  finished  in  pencil  lines.  In  this  case 
the  drawing  is  generally  little  more  than  a  rough  diagram,  hastily 
made,  and  not  intended  for  continued  use. 

Drawings  finished  in  ink  are  much  more  effective  and  desirable 
than  pencil  drawings ;  but,  as  a  good  inked  drawing  cannot  be  made 
except  upon  an  accurate  pencil  drawing,  pupils  should  begin  with 
the  pencil,  and  should  not  be  allowed  to  use  ink  until  they  can  pro- 
duce satisfactory  results  in  pencil. 

The  aim  in  pencil  work  should  be  to  approach  as  nearly  as  pos- 
sible the  accuracy  and  decision  which  are  only  attainable  in  an  ink 
drawing.  Care  in  selecting  and  using  pencils,  paper,  and  instru- 
ments will  produce  pencil  drawings  which  have  firm,  even,  black 
lines,  that  are  very  effective,  if  they  are  not  quite  so  strong  as  those 
made  with  ink. 

It  is  customary  to  represent  all  visible  edges  by  full  lines,  and  all 
invisible  edges  by  dotted  lines.  These  dotted  lines  should  be  as 
strong  as  the  full  lines  ;  they  should  be  regular  and  composed  not  of 
dots,  but  of  very  short  dashes,  whose  lengths  are  uniform  and  greater 
than  the  spaces  between  them.  Dotted  lines  are  often  very  poorly 
drawn,  and  spoil  the  effect  of  drawings  otherwise  good.  The  width 
of  the  line  and  the  length  of  the  dash  depend  upon  the  size  of 
the  drawing  and  its  purpose.  For  the  drawings  which  pupils  should 
make,  the  line  given  may  serve  as  a  copy. 


Care  must  be  taken  to  have  all  lines  stop  just  where  they  are 
intended  to  end.     If  lines  do  not  quite  meet,  or  if  they  pass  by  each 
other,  as  illustrated,  when  this  is  not  intended,   the 
drawing  is  spoiled. 

The  perfect  union  of  lines  which  should  be  tangent 
to  each  other  is  also  necessary  to  good  work.  Straight 
lines  should  run  into  arcs,  and  arcs  into  each  other,  without  the 
slightest  suggestion  of  a  break,  or  of  two  lines  instead  of  one. 

Centre  lines  are  necessary  in  working  drawings.     In  the  study  of 
projection,  it  is  not  necessary  to  show  them  unless  dimensions  are 


MA  TE RIALS  AND    THEIR    USES.  \  \ 

to  be  given.     They  are  represented  by  dot  and  dash  lines,  as  illus- 
trated, and  extend  some  slight  distance  outside  the  drawing,  to  show 


that  they  are  centre  lines.  Dimensions  are  placed  as  explained  in 
Art.  44. 

Different  materials  may  be  conventionally  shown  in  pencil  draw- 
ings by  using  different  kinds  of  section  lines.  The  materials  most 
commonly  used  are  often  represented  as  illustrated  in  Art.  76. 

An  accurate  and  neat  drawing  is  very  pleasing  and  effective ; 
every  draughtsman  and  advanced  student  should  be  able  to  produce 
such  a  drawing  with  ease.  Neatness  and  accuracy,  the  results  of 
care  and  long  practice,  are  essential  to  good  work  ;  and  pupils  should 
understand  that  without  the  exercise  of  much  patience,  work  of  real 
value  cannot  be  obtained. 


CHAPTER   11. 


GEOMETRICAL    PROBLEMS. 

THE  following  problems  are  those  most  likely  to  prove  valuable 
to  grammar  and  high  school  pupils.  They  are  given  in  order  that 
pupils  may  be  able  to  apply  them  in  the  study  of  design  and  working 
drawings,  and  that  they  may  learn  to  make  accurate  drawings. 
They  are  not  given  with  the  intention  of  teaching  geometry. 

The  illustrations  represent  the  working  lines  by  light  lines,  the 
given  lines  by  lines  of  medium  strength,  and  the  result  lines  by  heavy 
lines.  In  finishing  pencil  drawings  the  best  effects  are  obtained  by 
using  dotted  lines  for  the  working  lines  ;  but  if  time  necessary  to 
obtain  perfectly  regular  lines  cannot  be  given,  it  is  better  to  use  fine, 
light  lines.  When  drawings  are  inked  the  working  lines  may  be  red, 
the  given  lines  blue,  and  the  result  lines  black. 

In  working  alj  problems,  to  obtain  accurate  results,  arcs  de- 
scribed as  working  lines  should  be  of  large  radii. 

Problem  I.  —  To  bisect  a  straight  line  AB,  or 
an  arc  of  a  circle  ACB. 

With  A  and  B  as  centres  and  any  radius 
greater  than  half  AB,  describe  arcs  which  inter- 
sect at  i  and  2.  Join  i  and  2  by  a  straight 
line,  and  1-2  is  perpendicular  to  AB,  and 
bisects  it  in  j,  and  the  arc  in  C. 

Problem  2.  —  To  erect  a  perpendicular  to  a  given 
line  at  a  given  point  A  in  the  line. 

With  A  as  centre  and  any  radius,  set  off  equal 
distances  A I  and  A  2  from  A.  From  points  i  and  2 
as  centres,  and  with  a  radius  greater  than  half  1-2, 
describe  arcs  which  intersect  in  3  and  4.  Join  3  and 
4,  and  3-4.  is  the  required  perpendicular. 


GEOMETRICAL   PROBLEMS. 


Problem  3.  —  To  draw  a  perpendicular  to  a  given 
line  BC,from  a  point  A  otitside  the  line. 

With  A  as  centre  and  any  radius,  intersect  the 
'   given  line  in  points  I  and  2.     With  points  i  and  2 
as  centres  and  any  radius  describe  arcs  intersecting  in 
3.    Join  A  and  j ;  A  j  is  the  required  perpendicular. 

Problem   4.  —  To  erect  a  perpendicular  to  a  given  line  AB,  from 
a  point  B,  at  or  near  its  end. 

With  B  as  centre  and  any  radius,  draw  an  arc 
of  a  circle  1-2-3.  With  i  as  centre  and  the  same 
radius,  cut  this  arc  in  2,  and  with  2  as  centre  and 
the  same  radius,  describe  the  arc  3-4.  With  j  as 
centre  and  the  same  radius,  intersect  3-4  in  4. 
Join  4  B ;  this  is  the  required  perpendicular. 

Problem  5«  — To  draw  a  line,  EF,  parallel  to  a  given  line,  AB, 
and  the  distance  CD  from  it. 

From  any  two  points,  A  and  B,  in  the  line 
as  centres,  with  radius  CD,  describe  arcs  E 
and  F.  Erect  perpendiculars  at  A  and  B, 
which  intersect  the  arcs  in  E  and  F.  Join  E 
and  F;  this  is  the  required  parallel.  In  prac- 
tice, it  is  not  necessary  to  draw  the  perpen- 
diculars. 


Problem  6.  —  To  construct  an  equilateral  tri- 
angle on  a  given  base,  A  B. 

With  A  and  B  as  centres  and  A  B  as  radius, 
describe  arcs  which  intersect  at  i.  Join  i  A  and 
I  B. 

Problem  7.  —  To  construct  a  square  on  a 
given  base,  AB. 

Draw  A  i  perpendicular  to  AB  and  equal 
to  it.  (Problem  4.)  With  B  and  i  as  centres 
and  radius  AB,  describe  arcs  intersecting  in  2. 
Join  1-2  and  2B. 


MECHANICAL   DRAWING. 


Problem   8.  —  To  construct  a  rectangle  of  given  sides,   AB  and 
CD, 

At  A,  by  Problem  4,  erect  a  perpendicular 
Ai  equal  to  CD.  With  i  as  centre  and  radius 
AB,  describe  an  arc,  and  intersect  this  arc  in  2, 
by  one  described  from  B  with  CD  as  radius. 
Join  1-2  and  2B. 


Problem  9.  —  To  inscribe  a  square  within  a 
given  circle. 

Draw  AB,  a  diameter  of  the  circle,  which 
will  be  a  diagonal  of  the  square.  By  Problem 
i  bisect  AB,  and  continue  the  bisector  to  inter- 
sect the  circle  in  i  and  2.  Join  A  i,  i  B,  B  2, 
and  2  A. 


NOTE.  —  A  square  may  be  constructed  upon  a  given  diagonal  by  draw- 
ing a  circle  through  its  ends,  A  and  B,  and  proceeding  as  explained  above. 


Problem    10.  —  To  bisect  a  given  angle,  CAB. 

With  A  as  centre  and  any  radius,  describe  an 
arc  intersecting  AC  and  AB  in  points  i  and  2. 
With  points  i  and  2  as  centres  and  any  radius, 
describe  arcs  which  intersect  in  j.  Join  A  and  j. 

Problem    u.  —  To  trisect  a  right  angle,  CAB. 

With  A  as  centre  and  any  radius,  describe  an  arc 
intersecting  AC  in  i  and  AB  in  2.  With  i  and 
2  as  centres  and  the  same  radius  intersect  the  arc 
in  j  and  4.  Join  A  4  and  A  j. 

Problem  12. —  To  construct  at  E,  in  the  line  E  D, 
an  angle  equal  to  a  given  angle,  ABC. 

With  B  as  centre  and  any  radius,  intersect 
AB  and  BC  in  I  and  2.  With  E  as  centre  and 
same  radius  describe  an  arc  intersecting  E  D 
in  4.  With  4  as  centre  and  1-2  as  radius,  inter- 
sect the  arc  in  j.  Join  E  and  j. 


GEOMETRICAL   PROBLEMS. 


Problem    13.  —  To  construct  an  angle  of  po°,  60°,  45°,  30°,  15°, 
or  any  other  given  magnitude. 

There  are  360°  in  the  entire  circumference 
of  a  circle.  Draw  a  diameter  1-2,  and  there 
are  180°  on  either  side  of  this  line.  Draw  a 
second  diameter  3-4,  at  right  angles  to  1-2, 
and  the  circle  is  divided  into  four  equal  angles  of 
90°.  Trisect  one  of  these  by  Problem  n,  and 
angles  of  30°,  2c6,  and  of  60°,  2c$  are  ob- 
tained. Bisect  one  of  these  angles  of  30°,  and 
angles  of  15°,  2  c  7,  and  7  c  6  are  obtained.  Bisect  an  angle  of  90° 
and  angles  of  45°  are  obtained.  Trisect,  by  spacing,  an  angle  of  15° 
and  angles  of  5°  are  given.  .Divide  one  of  these  into  five  equal 
parts  and  degrees  are  given.  In  this  way  any  angle  may  be  obtained. 

Problem   14.  —  To  divide  a  given  line,  A  B,  into  any  number  of 
equal  parts,  as  five. 

Draw  AC  at  any  angle  to  AB.  On  AC  set 
off  any  distance  five  times.  Join  B  and  the  5th 
point,  and  through  the  other  points  draw  parallels 
to  B 5  (Art.  6),  which  will  divide  AB  as  required. 

Problem    15.  —  To  construct  a  triangle,  hav- 
ing given  its  sides,  AB,  C,  and  D. 

With  A  as  centre  and  radius  C,  describe  an 
arc.  With  B  as  centre  and  radius  D,  intersect 
this  arc  in  i.  Join  i  A  and  i  B. 

Problem  16.  —  To  inscribe  a  regular  hexagon  within  a  given  circle. 
Draw  the  diameter  AB  of  the  circle,  which 
will  be  a  long  diagonal  of  the  hexagon.     With 
point  A  as  centre  and  the  radius  of  the  circle 
as  radius,  intersect  the  circle  by  arcs  i  and  2. 
With  B  as  centre  and  same  radius,  intersect  the 
circle  by  arcs  3  and  4.      Join  A-i,  1-3,  3~B, 
B-4,  4-2,  and  2-A. 
NOTF. — By  joining  every  other  point  an  equilateral  triangle  will  be  obtained. 


i6 


MECHANICAL   DRAWING. 


Problem  17. —  To  construct  a  regular  hexagon 
upon  a  given  base  AB. 

With  A  and  B  as  centres  and  AB  as  radius, 
describe  arcs  which  intersect  at  i.  With  i  as 
centre  and  same  radius,  describe  a  circle,  and  in 
its  circumference  place  points  2,j,  4,  and  5,  which 
are  equidistant  and  the  distance  AB  apart.  Join 
A-2,  2-3, 3-4,  4-5,  and  $B. 

NOTE.  --  The  radius  of  any  circle  applied,  as  a  chord,  six  times  to  the 
circumference  divides  it  into  six  equal  parts. 


Problem  18.  —  To  construct  a  regular  octagon 
within  a  given  square  ABCD. 

Draw  the  diagonals  of  the  square,  which 
intersect  at  its  centre,  I,  and  with  points  A,  B, 
C,  and  D  as  centres  and  a  radius,  A  /,  equal  to 
half  the  diagonal,  describe  arcs  intersecting  the 
sides  of  the  square  in  points  2,  j,  8,  p,  6,  7,  4, 
and  j.  Join  4-9,  2-7,  8-5,  and  6-3. 


Problem  19.  —  To  inscribe  a  regular  pentagon 
within  a  given  circle  C. 

Draw  the  diameter  i— 2  and  a  radius  3-4  per- 
pendicular to  it.  Bisect  2—3  in  j',  and  with  j"  as 
centre  and  radius  5-4,  intersect  2-1  in  6.  With 
4  as  centre  and  radius  4-6  intersect  the  circle  in 
7.  Set  off  the  distance  4-7  from  7  to  S,  8  to  9, 
9  to  10.  Join  4-7,  7-8,  8-9,  p-/0,.and  10-4. 

Problem  2O.  —  To  draw  a  tangent  at  any 
point  A,  in  a  given  circumference. 

Draw  a  radius,  cA,  and  erect  a  perpen- 
dicular, 3-4,  to- the  radius  at  A,  by  Problem 
2.  This  is  the  required  tangent. 


GEOMETRICAL   PROBLEMS. 


Problem  21.  —  To  inscribe  a  circle  within  a  given 
triangle  ABC. 

By  Problem  10,  bisect  any  two  angles  of  the 
triangle,  as  CAB  and  ABC.  The  bisecting  lines 
intersect  at  i,  the  centre  of  the  required  circle.  A 
perpendicular,  as  1-2,  from  i  to  any  side  of  the  tri- 
angle, is  the  radius  of  the  required  circle. 

Problem  22.  —  To  draw  an  arc  of  a 
given  radius  EF,  tangent  to  given  straight 
lines  AB  and  CD. 

Draw  parallels  to  AB  and  CD  by  Prob- 
lem 5,  at  the  distance  .fi^from  them.  The 
parallels  intersect  at  i,  the  centre  of  the 
required  arc.  The  arc  is  tangent  to  AB 
and  CD,  at  points  3  and  2,  in  perpendiculars 
to  these  lines  from  i. 

Problem  23.  —  To  draw  an  arc  of  a  given  radius  DE,  tangent  to  a 
straight  line  AB  and  a  circle  C. 

Draw  by  Problem  5  a  parallel  P  3  to  AB, 
the  distance  DE  from  it.  With  i,  the  centre 
of  circle  C,  as  centre  and  a  radius  1-2,  equal 
to  the  radius  of  the  given  circle  plus  DE, 
that  of  the  required  arc,  describe  an  arc  to 
intersect  the  parallel  P 3  in  3,  the  centre  of 
the  required  arc.  A  perpendicular  from  3  to 
AB  gives  j",  the  point  of  contact  of  the  arc 

and  AB,  and  a  straight  line  from  3  to  i  gives  4,  the  point  of  contact 
of  the  arc  and  the  circle. 

Problem  24.  —  To  draw  an  arc  of  a  given 
radius  CD  tangent  to  tu>o  circles  A  and  B. 

With  i,  the  centre  of  the  circle  A,  as 
centre  and  radius  1-2,  equal  to  the  radius 
of  A  plus  CD,  describe  an  arc  2-3.  Inter- 
sect this  arc  in  5  by  one  described  from  j, 
the  centre  of  the  circle  B,  with  a  radius  3-4, 
equal  to  the  radius  of  B  plus  CD.  Point  5 


i8 


MECHANICAL   DRAWING. 


is  the  centre  of  the  required  arc,  and  lines  from  5  to  /  and  from 
5  to  3  give  6  and  7,  the  points  of  contact  of  the  arc  and  circles  A 
and  B. 


Problem  25.  — 


To  draw,  within  a  given  equilateral  triangle  ABC, 
three  equal  circles,  each  tangent  to  two 
others  and  to  one  side  of  the  triangle. 

Bisect  the  angles  of  the  triangle  and 
obtain  i.  Bisect  the  angle  I AB  and 
obtain  2,  the  centre  of  one  circle.  With 
i  as  centre  and  radius  1-2  draw  an  arc 
to  obtain  centres  j  and  4,  With  centres 
2,  3,  and  4  and  radius  2-j,  describe 
the  required  circles. 


Problem  26.  —  To  draw  arcs  of  circles  tangent  at  points  C  and  B, 
to  two  parallel  straight  lines  AB  and  CD  and  passing  through  any 

point  as  E,  in  the  line  CB. 

At  C  and  B  erect  perpendiculars. 
Bisect  BE  and  CE,  continuing  the  lines 
of  bisection  until  they  meet  the  perpen- 
diculars from  B  and  C  in  points  2  and  i, 
which  are  the  centres  of  the  required  arcs. 
With  i  as  centre  and  /  C  as  radius  draw 
the  arc  CE,  and  with  2  as  centre  and  2  B  as  radius  draw  the  arc  BE. 


Problem  27.  —  To  circumscribe  a  circle  about  a 
given  triangle  ABC. 

By  Problem  i  bisect  any  two  sides  as  AB 
and  BC  by  perpendiculars  meeting  at  /.  With 
i  as  centre  and  radius  /  A  describe  the  circle. 


NOTE  i.  —  If  any  three  points  as  A,  B,  C  not  in  the  same  straight 
line  are  given,  a  circle  may  be  passed  through  them  by  connecting  the 
points  and  proceeding  as  above. 

NOTE  2.  —  The  centre  of  any  circle  may  be  found  by  assuming  any 
three  points  in  it  and  proceeding  as  above. 


GEOMETRICAL    PROBLEMS.  19 

Problem  28.  —  To  construct  a  regular  octagon  on  a  given  side  AB. 

At  A  and  B  erect  perpendiculars  by 
Problem  4.  Continue  AB  and  bisect  the 
outer  right  angles,  making  the  bisecting 
lines  A  i  and  B  2  equal  to  AB.  Draw  1-2 
cutting  the  perpendiculars  in  3  and  4. 

From  3  and  4  set  off  the  distance  3-4, 
giving  j"  and  6.  Draw  through  5  and  6  a 
straight  line,  and  set  off  from  j"  and  6  dis- 
tances 5-7  and  5-8,  6-9  and  <5-^o  equal 
to  3-1  or  j>  A.  Join  the  points  7-7,  7-8,  8-9,  9-10,  and  70-2. 


Problem  29.  —  72?  inscribe,  within  a  given  equilateral  triangle 
ABC,  three  equal  circles.,  each  tangent  to  two  others  and  to  two  sides  of 
the  triangle. 

Bisect  the  angles  A,  B,  and  C  by  lines 
meeting  at  i  •  bisect  the  right  angle  C  2  B 
to  obtain  3,  the  centre  of  one  of  the  required 
circles.  With  i  as  centre  and  1-3  as  radius, 
describe  a  circle  giving  points  4  and  5. 
3,  4,  and  5  are  the  centres  of  the  required 
circles.  The  shortest  distance,  as  3-6,  from 
a  centre  to  a  side  of  the  triangle,  is  the 
radius  of  the  required  circles. 


Problem  30.  —  Within  a  given  square, 
A  BCD,  to  draw  four  equal  circles,  each 
touching  two  others  and  two  sides  of  the 
square. 

Draw  the  diagonals  of  the  square  to 
obtain  its  centre,  and  then  lines  parallel 
to  the  sides  through  the  centre  to  inter- 
sect the  sides  in  points  i,  2,  3,  and  4. 
Join  1-3,  3-2,  2-4,  and  4-1,  obtaining 
points  j,  6,  7,  and  8,  the  centres  of  the 
required  circles.  The  length,  7-9,  of  a  radius  is  given  by  drawing  7-8. 


2O 


MECHANICAL   DRA  WING. 


Problem  31.  —  To  divide  a  straight  line, 
CD,  into  the  same  proportional  parts  as  a 
given  divided  line,  AB. 

Draw  CD  parallel  to  AB  by  Problem  5. 
Draw  lines  through  AC  and  BD  to  meet  in 
/.  From  points  a,  b,  and  c  in  AB,  draw  to 
i  lines,  which  will  divide  CD  as  required. 

Problem  32.  —  Within  a  square,  ABCD, 
to  draw  four  equal  semi-circles,  each  touch- 
ing one  side,  and  forming  by  their  diameters  a 
square. 

Draw  the  diagonals  and  the  diameters 
of  the  square  and  join  points  1-2,  2-3, 3-4, 
and  4-1,  obtaining  3,  6,  7,  and  8.  Join  5-6, 
6-7,  7-8,  and  8-5,  obtaining  p,  10,  n,  and 
12,  which  are  the  centres  of  the  required 
arcs,  of  which  9-2  is  the  length  of  a  radius. 

-  Problem  33.  — -To  construct  an  equilat- 
eral triangle,  its  altitude,  AB,  being  given. 

At  A  and  B  erect  perpendiculars  to  AB 
by  Problem  4.  With  A  as  centre  and  any 
radius  describe  a  semi-circle,  which  inter- 
sects the  perpendicular  in  /  and  2,  and  from 
/  and  2  set  off  the  radius  of  the  semi-circle 
in  3  and  4.  Draw  A  3  and  A  4  to  intersect, 
in  3  and  6,  the  perpendicular  erected  at  B. 

Problem  34.— 70  circumscribe  an  equi- 
lateral triangle  about  a  given  circle  C. 

Draw  a  diameter,  r-2,ot  the  circle.  With 
2  as  centre  and  2-3,  the  radius  of  C,  as 
radius,  describe  a  circle  intersecting  1-2 
produced  in  4  and  the  circle  in  3  and  6. 
With  i,  3,  and  6  as  centres,  and  radius  1-3, 
describe  arcs  which  intersect  at  7  and  8. 
Join  4-7,  7-8,  and  8-4. 


GEOMETRICAL    PROBLEMS. 


21 


cn 

v^ 

5 

N.                          \ 

{^A 

Problem  35.  —  2o  circumscribe  a  square 
about  a  given  circle. 

Draw  the  diameter  1-2,  and  a  diameter 
3-4,  at  right  angles  to  1-2.  With  i,  2,  3, 
and  4.  as  centres  and  radius  7-5,  describe 
arcs  which  intersect  at  6,  /,  8,  and  p.  Join 
,  6-7,  7-8,  and  <i?-p. 


Problem   36.  —  To  circumscribe  a  regular  hexagon  about  a  given 

circle  C. 

Draw  a  diameter,  1-2,  and  with  the  radius 
of  the  circle  set  off,  from  i  and  2,  points  3,  4, 
5,  and  6 ;  through  these  points  draw  radii,  ex- 
tending them  beyond  the  circle.  Bisect  the 
sector  i  c  3  by  c  f,  which  intersects  the  circle 
at  8.  At  8,  by  Problem  2,  erect  a  perpendicular 
and  obtain  9  and  7^.  With  c  as  centre  and  c  p 
as  radius,  describe  a  circle  to  intersect  the 
radii  extended  in  70.  77.  12,  and  7j,  the  vertices 

of  the  required  hexagon.    Join  p-70,  70-77,  11-12,  12-13,  and  13-14. 

Problem    37.  —  7i>  draw  circles  tangent  to  each  other  and  to  two 

lines,  AC  and  BC,  not  parallel,  the  raditis,  DE,  of  one  circle  being 

given. 

Bisect  the  angle  ACB  by  C  i.  Draw 
parallel  to  AC,  and  distant  from  it  by  the 
radius  of  the  given  circle,  a  line  which,  in- 
tersecting C  i  at  2,  gives  the  centre  of 
the  first  circle,  which  intersects  C  i  in  j- 
At  j  erect  a  perpendicular  to  C  7  and 
obtain  4,  and  bisect  the  angle  j-^-C, 
obtaining  5,  the  centre  of  the  second 

circle,  whose  radius  is  the  distance  5-3.     In  this  way  any  number  of 

circles  may  be  drawn. 

Problem  38.  —  To  construct  a  regular  polygon  of  any  number  of 
sides  (in  this  case  seveti),  upon  a  given  base,  AB. 


22 


MECHANICAL    DRA  WING. 


Extend  BA,  and  with  A  as  centre  and 
AB  as  radius  describe  a  semi-circle.  Divide 
the  semi-circle  into  as  many  equal  parts  as 
there  are  sides  in  the  required  polygon,  and 
draw  from  A  to  the  second  point,  a  second 
side  of  the  polygon.  Bisect  A  2  and  AB ; 
the  bisectors  meet  at  7,  the  centre  of  the 
required  polygon.  With  7  as  centre  and 
7  A  as  radius,  describe  a  circle  and  upon  it 
set  off  the  distance  AB,  beginning  at  2,  and 

obtain  points  8,  9,  10,  and  iz,  the  vertices  of  the  required  polygon. 

Join  2-8,  8-9,  9-10,  10-11,  and  n  B. 

Problem  39.  — =  To  inscribe  a   regular  polygon  of  any    number  of 
sides  (in  this  case  five)  within  a  given  circle. 

Draw  a  diameter  AB  and  divide  it 
into  as  many  equal  parts  as  the  poly- 
gon is  to  have  sides.  With  A  and  B  as 
centres  and  radius  AB,  describe  arcs 
intersecting  in  5.  From  5  draw  a  straight 
line  through  the  second  point,  2,  to  inter- 
sect the  circle  in  6.  A  6  is  a  side  of  the 
required  polygon.  Beginning  at  6,  set 
off  A  6  upon  the  circle  to  obtain  points  7,  8,  9.  Join  A  6,  6-7 
7-8,  8-9,  and  9  A. 

NOTE.  —  This  method  is  approximate  only. 

Problem    40.  —  Within    a  given    circle  to  draw   any  number  of 
equal  semi-circles  tangent  to  the  circle  and  forming  by  their  diameters  a 

regular  polygon. 

Draw  radii  1-2  and  i-j  at  right  angles 
to  each  other.  Beginning  at  2,  divide  the  circle 
into  twice  as  many  equal  parts  as  the  number 
of  semi-circles  required,  and  draw  diameters  to 
these  points.  Join  2  and  j  ;  the  intersection  9 
of  2—3  with  the  diameter  4-7  is  a  vertex  of  the 
polygon  whose  sides  are  the  diameters  of  the 
required  semi-circles.  Describe  a  circle  through 


GEOMETRICAL   PROBLEMS. 


p  with  i  as  centre,  and  join  its  points  of  intersection,  g,  10,  and 
n,  obtaining  the  diameters  which  contain  points  12,  ij,  and  14, 
the  centres  of  the  required  semi-circles,  of  which  12-2  is  the 
length  of  a  radius. 

Problem    41. —  Within  a  given  circle  A  to  draw  any  number  of 

equal  circles  tangent  to  each  other  and  to  the  circle  A. 

Divide  the  circumference  of  the  cir- 
cle into  as  many  equal  parts  as  inscribed 
circles  are  required,  and  draw  radii 
to  these  points  of  division.  Bisect  the 
angle  i  c  5,  and  at  5  draw  a  tangent 
which  intersects  the  bisector  at  6.  Bisect 
the  angle  5~6-c  and  obtain  7,  the  centre 
of  one  of  the  required  circles.  With  c  as 
centre  and  c  7  as  radius,  describe  a  circle 

to  cut  the  radii  previously  drawn,  in  points  8,  9,  10,  and  n,  which 

are  centres  of  the  required  circles.     The  radius  of  each  circle  is  the 

distance  7-5. 

Problem    42 .  —  About  a  given  circle  A,  to  draw  any  number  of 
equal  circles  tangent  to  each  other  and  to  the  given  circle. 

Divide  the  circumference  of  the  circle 
into  twice  as  many  equal  parts  as  the  number 
of  circles  required,  and  draw  radii  extend- 
ing them  beyond  the  points  of  division.  At 
any  point,  as  2,  draw  a  tangent  to  the  circle 
intersecting  a  radius,  extended,  at  n.  Bisect 
the  angle  2-11-12,  and  obtain  ij,  the  centre 
of  one  of  the  required  circles.  With  c  as 
centre  and  c-ij  as  radius,  describe  a  circle 

to  intersect  every  other  radius  in  14,  15,  16,  and  77.  These  points 
are  the  centres  of  the  required  circles  of  which  the  radius  is  the 
distance  13-2. 

Problem   43.  —  To  draw  an  ellipse,   its  axes  AB  and  CD  being 

given. 


MECHANICAL    DRAWING. 


Upon  the  axes  draw  circles  and  divide 
both  circles  by  diameters  i-i  •  2-2,  etc., 
into  any  number  of  parts,  equal  or  un- 
equal. From  the  points  in  the  large  circle 
draw  parallels  to  the  short  axis,  and  from 
those  in  the  small  circle  draw  parallels  to 
the  long  axis.  The  intersections  of  lines 
from  points  of  the  same  number  are  points 
in  the  ellipse. 

Problem   44.  —  To  draw,  by  means  of  a  trammel,  an  ellipse,  whose 

axes  are  given. 

Set  off  AB,  half  the  long  axis,  on  the 
edge  of  a  straight  piece  of  paper  from  P 
to  S,  and  AC,  half  the  short  axis,  from  P 
to  L.  Move  this  paper  trammel  so  that 
S  is  upon  the  short  axis  and  L  upon  the 
long  axis,  and  P  will  always  be  a  point  in 
the  ellipse.  Its  position  may  be  marked 
by  a  sharp  lead  pencil.  The  best  way  is 
to  use  a  needle  point,  and  make  very  fine 

punctures  in  a  straight  line  drawn  near  the  edge  of  the  trammel. 

The  needle  may  then  be  placed  through  L  and  on  the  long  axis,  and 

the  paper  revolved  about  L  till  the  point  S  is  over  the  short  axis, 

when  the  needle  may  be  used  to  mark,  through  P,  a  fine  puncture 

in  the  paper. 

To  enable  the  axes  to  be  seen,  the  edges  of  the  paper  may  be 

notched  as  indicated. 


Problem  45. 

proximate  ellipse. 


—  To  draw  upon  given  axes,  AB  and  CD,  an 


With  the  centre,  6,  of  the  ellipse  as 
centre  and  half  the  short  axis,  6  C ,  as 
radius  describe  an  arc  C  i.  Draw  CB  and 
from  C  set  off  C  2  equal  to  B  i,  or  the 
difference  between  half  the  short  and  half 
the  long  axis.  Bisect  B  2  and  continue 
the  bisecting  line  to  intersect  AB  in  j, 


GEOMETRICAL    PROBLEMS. 


and  CD  in  4.  With  3  as  centre  and  radius  j  13  describe  an  arc 
/>'  5,  and  with  4  as  centre  and  4-5  as  radius,  describe  an  arc  which 
will  pass  through  C  and  complete  one  quarter  of  the  curve.  From 
6,  set  off  6-7  equal  to  6-3,  and  6-8  equal  to  6-4,  and  draw  lines 
joining  ^-7,  8-7,  and  8—3,  and  corresponding  to  4.-$.  With 
points  3,  4,  7,  and  8,  as  centres,  complete  the  ellipse  as  ex- 
plained above. 

Problem    46.  —  To  draw  an  equable  spiral. 

Draw  AB  and  upon  it  place  any  two  points, 
i  and  2,  as  centres.  With  i  as  centre,  and 
radius  1—2,  draw  a  semi-circle.  With  2  as 
centre  and  radius  2—3,  draw  a  semi-circle ; 
continue  the  process,  using  i  and  2  as  centres, 
and  the  distance  to  the  end  of  the  diameter  of 
the  semi-circle  last  drawn  as  radius. 

Problem   47.  —  To  draw  a  variable  spiral,  its  greatest  diameter, 

AB,  being  given. 

Divide  AB  into  eight  equal  parts  by 
points  a,  b,  c,  etc.  On  d  e,  the  fifth  space, 
as  diameter,  draw  a  circle  which  is  called 
the  eye  of  the  spiral.  Inscribe  a  square  in 
the  circle  and  draw  its  diameters,  as  shown 
in  the  enlarged  drawing  of  the  eye.  Divide 
these  diameters  into  six  equal  parts  and 
number  the  points  as  shown.  W7ith  i  as 
centre  and  1-13  as  radius,  draw  the  arc 
13-14.  With  2  as  centre  and  2—14,  as 
radius,  draw  the  arc  14-15.  With  3  as 
centre  and  3-15  as  radius,  draw  the  arc 

1^-16 ;  and  so  proceed,  using  all  the  points  up  to  and  including  12, 

as  centres^  and  limiting  each  arc  by  a  line  drawn  from  its  centre 

through  the  centre  of  the  next  arc. 


26 


MECHANICAL    DRA  WING. 


Problem    48.  —  To  draw  the  involute  of  a  circle. 

From  any  point,  as  7,  in  the  circum- 
ference set  off  any  number  of  equidistant 
points,  2,  j,  4,  and  5.  Draw  tangents  to 
the  circle  at  these  points,  and  on  the  tan- 
gent at  2  set  off  the  distance  2-1,  giving  6. 
On  the  tangent  at  j,  set  off  the  length  of 
the  arc  j-7,  giving  8.  At  4  set  off  the 
length  of  the  arc  4-1,  giving  77.  At  j  set 
off  the  length  of  the  arc  5-7,  giving  75. 
A  curve  through  7,  6,  #,  77,  and  75  is 
the  involute  of  the  circle. 

NOTE.  —  The  distance  between  points  i,  2,  3,  etc.  must  be  such  that 
the  difference  in  length  between  the  chord  and  the  arc  is  slight. 

Problem   49.  —  To  draw  the  cycloid  curve  traced  by  a  point  in  tJic 
circumference  of  a  given  circle,  A,  which  rolls  on  the  line  CD. 

Divide  the  circumference  into 
any  number  of  equal  parts,  as 
twelve,  and  set  off  on  CD  the 
equidistant  points  in  which  the 
points  marked  in  the  circle  will 
come  to  the  straight  line  as  the 
circle  rolls  upon  CD.  When  the 
circle  is  tangent  at  2'  its  centre 
will  be  in  a  perpendicular  erected 

at  2'  and  in  a  parallel  to  CD  through  o,  the  centre  of  A.  In  the 
same  way  its  centre  will  be  found  when  it  is  tangent  at  3',  4',  5',  etc. 
As  the  circle  rolls,  point  i  describes  the  cycloid.  In  any  position 
of  the  circle  point  i  will  be  found  by  setting  off  as  many  equal 
spaces  on  the  circle  from  points  2\  3  ',  4 ',  etc.,  as  the  circle  has  rolled 
over,  starting  from  7'.  Thus,  one  space  from  2'  to  a  ;.  two  spaces 
from  j'  to  b,  and  so  on,  for  the  complete  curve.  Points  a,  b,  c,  etc., 
may  also  be  found  by  noting  the  level  of  the  points  6,  5,  4,  etc.,  in 
the  given  circle  A.  Thus,  a  horizontal  line  from  6  will  give  a  ;  one 
from  5  will  give  b,  and  so  on.  In  either  of  these  ways,  the  complete 


GEOMETRICAL    PROBLEMS. 


curve,  of  which  one-half  only  is  shown,  may  be  obtained.  The 
length  of  the  arc  between  the  points  in  the  circle  of  this  and  Prob. 
48  may  be  computed,  or  a  result  practically  correct  obtained  by 
dividing  the  circle  into  many  parts. 

Problem  50. —  To  construct  the  epicycloid  curve  traced  by  a  point 
in  the  circumference  of  a  circle,  A,  which  rolls  on  the  outside  of  a  circle  B. 

Divide  the  circumference  of  the 
circle  A  into  any  number  of  equal 
parts,  as  twelve,  and  set  off  from  7 
on  B  the  points  6',  5',  4',  j',  2',  i\ 
in  which  the  points  6,  5,  4,  j,  2,  i 
of  A  will  coincide  with  B  as  A  rolls. 
Describe  an  arc  of  A  when  it  is  tan- 
gent, at  each  of  the  points  6\  5',  ^', 
j',  2' ;  the  centres  will  be  in  an  arc 
" concentric  with  B,  passing  through 
<?,  and  in  radii  of  B,  extended 
through  the  points  6',  jj',  4',  j',  and 
2'.  When  tangent  at  2'  the  marking 
point  i  is  at  a,  and  is  obtained  by 
setting  off  from  2'  one  of  the  equal 
spaces  into  which  it  was  divided,  or 

by  drawing  an  arc  through  6,  with  D  as  centre,  to  intersect  at  a  the 
circle  A,  when  tangent  at  2'.  When  tangent  at  j'  the  marking  point 
i  will  be  at  b,  and  will  be  obtained  by  setting  off  two  equal  spaces 
from  j'  to  b,  or  by  an  arc  through  5.  In  this  way  all  the  points,  c, 
d,  e,  of  the  curve  may  be  found ;  also  the  other  half  of  the  curve. 

Problem  51.  —  To  construct  the  hypocycloid  traced  by  a  .point  in 
circle  C,  which  rolls  inside  the  circle  B. 

The  process  is  the  same  as  for  the  outer  epicycloid  explained  in 
Problem  50. 

NOTE.  —  When  the  diameter  of  the  rolling  circle  is  equal  to  the  radius  of 
the  circle  within  which  it  rolls,  the  hypocycloid  becomes  a  straight  line  and 
is  a  diameter  of  B. 


CHAPTER   III. 

WORKING    DRAWINGS. 

13.  Nature  and  use  of  Working  Drawings. — The   workman 
who  builds  a  house,  engine,  or  any  other  object,  obtains  the  neces- 
sary information  concerning  the  size,  shape,  kind  of  material,  and 
amount  of  finish,  etc.,  of  all  the  different  parts,  from  different  projec- 
tions or  views  of  the  object  to  be  made.     These  views  are  called 
working  drawings. 

14.  Free-hand  drawings,  that  is,  perspective  views,  give  the  appear- 
ance of  objects,  but  they  represent  only  part  of  the  surface  of  an 
object,  and  do  not  give  the  actual  sizes  or  shapes  of  any  of  its  parts  ; 
therefore  such  drawings  are  not  suitable  for  working  drawings. 

A  perspective  drawing  may  be  used  as  a  working  drawing  when 
the  object  illustrated  is  very  simple,  or  of  a  form  commonly  used,  and 
the  dimensions  of  the  parts  are  placed  upon  the  drawing ;  but  when 
such  a  drawing  is  used  the  knowledge  and  experience  of  the  work- 
man supply  the  information  which  the  perspective  drawing  does  not 
give. 

Generally,  perspective  views  of  machinery  and  architecture  are 
made  simply  as  illustrations  of  objects  which  either  have  not  been 
made,  or  which  are  not  placed  so  they  can  be  photographed.  The 
best  and  cheapest  perspective  possible  to  obtain  is  given  by  a  photo- 
graph. 

!$•  A  working  drawing  must  give  the  actual  shapes  of  the  parts 
it  represents.  It  may  be  the  full  size  of  the  object  or  smaller,  but  it 
must  be  drawn  to  some  determined  scale  and  show  the  true  propor- 
tions and  relations  of  all  the  parts.  In  projection  the  actual  appear- 
ance of  any  object  is  given  upon  a  plane  at  right  angles  to  the  direc- 
tion in  which  the  object  is  seen.  It  follows  that,  as  far  as  possible, 
working  drawings  should  be  made  upon  planes  which  are  parallel  to 
the  principal  faces  of  the  object  to  be  represented. 


WORKING  DRAWINGS. 


29 


16.  The  nature  and  relations  of  the  different  views  usually  made 
as  working  drawings  will  be  shown  by  the  following  experiments  : 

Place  a  cube,  shown 
by  Fig.  i,  at  the  level 
of  the  eye  so  that  only 
one  face  is  visible,  the 
corners  of  this  face 
being  equally  distant 
from  the  eye.  Hold  a 
glass  slate  vertically  in 
front  of  the  cube  and  FlG-  I- 

parallel  to  the  visible  face,  that  is,  at  right  angles  to  the  direction  in 
which  the  cube  is  seen.  Trace  the  appearance  of  the  cube  upon  the 
slate.  The  tracing  will  be  a  square,  for  the  apparent  shape  of  any 
surface  at  right  angles  to  the  direction  in  which  it  is  seen  agrees  with 
its  real  shape. 

If  the  slate  is  now  held  horizontally  above  the  top  of  the 
cube  and  the  eye  is  placed  above  the  object,  so  that  the  top  face 
appears  its  real  shape,  a  tracing  of  the  cube  from  this  position  will 
be  a  square,  as  in  the  first  case.  The  first  tracing  gives  the  real 
shape  of  the  front  face  of  the  cube,  and  the  second  tracing  the  real 
shape  of  the  top  face  of  the  cube.  The  tracing  on  the  slate  when 
vertical  shows  that  the  height  of  the  object  is  equal  to  its  width, 
while  that  upon  the  slate  when  horizontal  shows  that  the  distance 
from  front  to  back  is  equal  to  the  width.  The  two  tracings  together 
express  the  fact  that  the  three  dimensions  of  the  cube  are  equal. 
This  will  be  shown  whatever  the  distances  of  the  slate  and  the  eye 
from  the  cube ;  but  to  have  the  two  tracings  of  the  same  size,  the 
eye  and  slate  must  have  the  same  relative  positions  with  reference 
to  the  cube  when  making  both. 

17.  The  tracings  are  smaller  than  the  cube  because  the  visual 
rays  converge  toward  the  eye,  and  the  slate  is  nearer  the  eye  than  is 
the  cube.     If  the  eye  is  placed  farther  away  from  the  object,  and  the 
position  of  the  slate  remains  the  same,  the  tracing  will  be  larger.     If, 
as  the  tracing  is  made,  the  eye  is  moved  so  as  to  see  every  visible 
point  of  the  object  by  means  of  a  visual  ray  perpendicular  to  the 
slate,  the  tracing  will  be  the  full  size  of  the  object. 


30  MECHANICAL    DRAWING. 

18.  If  instead  of  one  slate  which  is  held  in  the  hand  to  receive 
the  different  views  of  the  cube,  two  slates  are  used  which  are  sup- 
ported at  right  angles  to  each  other,  as  in  Fig.  2,  and  if  instead  of 
tracing  the  appearance  when  the  eye  is  in  one  position,  the  eye  is 
moved  with  the  pencil  point  so  that  every  point  of  the  object  is  seen 
in  a  perpendicular  to  the  slate,  the  tracing  upon  the  vertical  slate 
will  give  the  real  dimensions  of  the  front  face  of  the  cube,  and  that 
upon  the  horizontal  slate  will  give  the  real  dimensions  of  the  top 
face  of  the  cube.     The  two  tracings  will  give  all  the  actual  dimen- 
sions of  the  object,  and  the  dimensions  of  one  tracing  may  be  com- 
pared with  those  of  the  other.     These  tracings  illustrate  the  nature 
of  constructive  drawings,  and  the  slates  in  their  different  positions 
illustrate  the  imaginary  planes  upon  which  these  drawings  are  sup- 
posed to  be  made. 

19.  These   tracings   or   drawings   are   called   "projections"    or 
"views,"  and  when  dimensioned  they  are  called  working  drawings. 
The  tracing  upon  the  vertical  slate  is  called  the  vertical  projection, 
or   the   front  view;    that  upon  the  horizontal   slate   is   called  the 
horizontal   projection,    or   the    top   view.       Working   drawings    are 
generally  called  views,  which  will  be  the  term  used  throughout  this 
book. 

The  pupil  who  has  used  the  transparent  drawing  slate  for  free- 
hand purposes  will  have  no  difficulty  in  making  and  understanding 
these  experiments ;  he  will  readily  see  that  the  tracings  made  upon 
the  vertical  and  horizontal  slates  give  the  three  actual  dimensions  of 
the  cube  seen  through  them. 

20.  These  drawings  might  be  used  to  form  the  cube  represented, 
by  so  shaping  any  material  that,  when  it  is  looked  at  horizontally, 
it  will  cover  the  tracing  upon  the  vertical  slate,  and  when  it  is  looked 
at  from  above,  it  will  cover  that  upon  the  horizontal  slate. 

Objects  are  sometimes  shaped  by  cutting  them  until  they  exactly 
cover  the  drawing  which  represents  them  :  this  is  the  case  particu- 
larly with  irregularly  curved  parts.  Such  parts  are,  however,  gener- 
ally obtained  by  the  use  of  templets,  which  are  forms  of  thin  wood 
or  metal  made  from  the  drawings,  and  applied  to  the  object  to  show 
when  the  desired  form  has  been  obtained.  In  this  way  the  frames 
of  boats  are  shaped ;  also  other  irregularly  curved  objects. 


WORKING   DRAWINGS.  31 

21.  Objects  are  seldom  shaped  by  placing  them  upon  the  draw- 
ings, since  the  drawings  are  often  less  than  the  full  size  of  the  object. 
Even  when  this  is  not  the  case  it  is  difficult  to  obtain  accurate  work 
in  this  way. 

22.  Drawings  which  are  intended  for  shop  use  always  give  in 
figures  the  dimensions  of  each  part ;  also  directions  as  to  the  material 
of  which  it  is  to  be  made,  and  the  amount  of  finish  which  it  is  to 
receive.     The    machinist  readily  works  to   hundredths    of  inches ; 
accuracy  such  as  this,  or  even  far  less,  would  be  impossible  by  such 
a  clumsy  method  as  placing  the  object  upon  the  drawing. 

23.  The  dimensioning  of  the  drawings  is  to  the  workman  of  first 
importance,  as  he  is  not  allowed  to  apply  a  scale  to  obtain  a  dimen- 
sion omitted  from  the  drawing  ;  but  to  the  student  it  is  not  at  first 
important.     His  attention  should  be  given  to  the  study  of  the  princi- 
ples necessary  to  enable  him  to  make  drawings.     Wh^n  able  to  do 
this  he  should  study  dimensioning,  which  is  advisable  in  the  seventh 
and  eighth  grades. 

24.  Drawings  are  made  upon  paper,  which  cannot  be  used  as 
was  the  glass  slate  to  trace   the   appearance  of  an  object ;   but  as 
practical  working  drawings  are  not  made  by  tracing,  and  since  the 
chief  use  of  a  working  drawing  is  to  represent  what  exists  only  in 
the  brain  of  the  engineer,  architect,  or  designer,  working  drawings 
would  be  of  little  service  if  they  could  be  obtained  only  by  tracing  or 
by  drawing  from  objects  actually  existing. 

Drawings  represent  so  exactly  the  conceptions  of  the  designer 
that  the  greatest  mechanical  constructions  are  made  in  small  sections 
in  the  workshop,  and  are  then  taken  away  and  put  together,  every 
screw  and  bolt  hole  of  every  part  being  properly  placed,  so  that  the 
"  Ferris  Wheel,"  or  the  engine  of  the  steamship  "  Paris,"  or  even  the 
entire  boat  with  all  her  fittings,  can  be  put  together  the  first  time  as 
surely  as  can  the  parts  of  a  simple  box. 

The  following  paragraph  from  the  Century  for  July,  1894,  gives 
an  idea  of  the  importance  of  the  draughtsman's  duties. 

"  For  the  hull  alone  of  the  battleship  Indiana,  25  principal  plans  must 
be  made,  and  fully  400  separate  drawings  must  be  prepared,  and  duplicated 
by  photographs.  This  of  itself  is  enough  work  to  keep  a  force  of  expert 
draftsmen  busy  continuously  for  eight  months.  For  the  engines  more  than 


32  MECHANICAL    DRAWING. 

250  separate  drawings  are  required,  and  these  in  all  their  intricate  details 
would  take  a  force  of  50  men  nearly  a  year  to  complete,  if  engaged  continu- 
ously at  the  task.  Not  only  must  every  rivet  and  every  joint  be  marked  out 
and  noted,  but  there  must  be  the  most  complicated  computation  of  strains 
and  weights." 

25.  Study  of  Principles.  —  In  order  to  understand  readily  the 
principles  of  the  subject  of  working  drawings,  two  slates  should  be 
hinged  together  so  that  they  can  be  placed  at  right  angles  to  each 
other,  and  can  be  revolved  into  the  same  plane,  as  shown  by  Figs.  2, 
3,  and  4. 

Fig.  2  represents  a  cube  and  the  vertical  and  horizontal  glass 
slates,  which  represent  the  imaginary  planes  upon  which  the  front 


/* 

T 

. 

F«OI»T    PlAfl 

r 

f 

. 

f 

FIG.  2. 


FIG.  3. 


FIG.  4. 


and  top  views  are  supposed  to  be  made  ;  Fig.  3  represents  the  slates 
after  the  horizontal  one  has  been  revolved  to  coincide  with  the  plane 
of  the  vertical  slate  ;  Fig.  4  is  a  drawing  which  gives  the  real  shapes 
•of  the  two  slates  and  of  the  views  upon  them. 

26.  It  will  be  seen  that  after  the  slates  have  been  placed  as  illus- 
trated by  Figs.  3  and  4,  the  views  will  be  in  line  with  each  other,  the 
top  view  being  above  the  front  view,  and  the  two  views  of  any  point 
being  in  the  same  vertical  line. 

The  drawings  also  show  a  vertical  line  1-2,  and  its  views  upon  the 
two  slates. 


WORKING  DRAWINGS. 


33 


27.  It  often  happens  that  these  two  views  are  not  enough  to  de- 
scribe an  object.  Thus,  if  the  cube  is  bisected  to  form  two  triangu- 
lar blocks,  the  views  of  either  triangular  block  upon  the  vertical  and 
horizontal  slates  will  be  the  same  as  the  views  of  the  entire  cube. 

Fig.  5  represents  a  triangular  block  formed  by  bisecting  the  cube, 
illustrated  in  Fig.  2,  by  a  plane  passing  through  the  front  upper  and 


FIG.  5. 


FIG.  6. 


LEFT 


RIGHT 


FIG.  7. 


back  lower  edges ;  in  Fig.  5  these  edges  are  2-3  and  1-4  respec- 
tively. It  also  represents  side  slates  attached  to  the  vertical  and 
horizontal  slates,  and  at  right  angles  to  both.  .If  a  tracing  of  the 
block  is  made  upon  one  or  both  of  these  side  planes,  or  slates,  as 


34 


MECHANICAL   DRAWING. 


upon  the  front  and  top  slates,  the  information  not  given  by  the  first 
two  views  will  be  supplied. 

Figs.  6  and  7  represent  the  slates  when  the  side  slates  and  the  top 
slate  have  been  revolved  to  coincide  with  the  plane  of  the  front  slate. 

28.  The  views  upon  the  side  slates  are  at  the  same  level  as  that 
upon  the  front  slate,  and  when  the  slates  are  as  shown  in  Figs.  6 
and  7,  the  right  and  left  side  views  of  any  point  are  in  a  horizontal 
line  through  the  front  view  of  the  point. 

29.  A  fifth  slate  may  be  placed  at  the  back,  and  thus  a  view  of 
the  back  may  be  obtained.     The  surface  upon  which  the  object  rests 
may  be  that  of  a  sixth  slate,  and  thus  a  view  of  the  bottom  may  be 
traced.     All  these,  together  with   additional   views,  are   sometimes 
necessary.     This  arrangement  of  slates  and  object  really  amounts  to 
placing  about  the  object  a  glass  box  and  tracing  upon  the  sides  of 
the  box,  by  means  of   perpendiculars  to    the   sides,   the  different 
appearances  of  the  object. 

30.  These  perpendiculars,  or  visual  rays,  which  produce  the  dif- 
ferent  views    are    called   projecti?ig  lines.      When    the    planes    are 
revolved  to  coincide  with  the  plane  of  the  front  plane,  as  shown  by 
Figs.  6  and  7,  these  projecting  lines  are  represented  by  the  vertical 
and  horizontal  lines  which  contain  the  different  views  of  the  various 
points,  and  are  the  projecting,  or  working,  lines  of  the  drawings. 

31.  The  figures  show  that  the  different  views  are  arranged  with 
reference  to  the  front  view,  so  that  the  line  of  each  view  nearest  the  front 
view  represents  the  front  face  of  the  object. 

By  means  of  such  glass  slates  hinged  together,  these  simple  prin- 
ciples may  be  so  illustrated  that  pupils  in  the  upper  grammar  grades 
can  understand  the  nature  of  the  drawings  made  for  the  different 
views  of  the  various  objects  chosen  for  study,  the  reason  for  arrang-' 
ing  working  drawings  as  explained  ;  also  the  fact  that  the  front  and 
top  views  of  the  same  point  are  in  the  same  vertical  line,  and  the 
front  and  side  views  in  the  same  horizontal  line. 

32.  Some  teachers  prefer  to  think  of  the  object  as  revolved  so 
that  its  different  surfaces  are,  one  after  another,  placed  parallel  to 
one  plane  (or  glass  slate)  upon  which  the  drawings  are  made  in  the 
same  manner  as  a  drawing  made  upon  any  one  of  the  slates  at  right 
angles  to  each  other.     There  is  no  objection  to  this  method  ;  indeed 


WORKING   DRAWINGS. 


35 


in  the  practical  work  of  making  drawings  from  the  object  it  is  really 
always  done.  The  slates  at  right  angles  to"  each  other  are  used 
simply  to  illustrate  the  principles  governing  the  making  and  arrange- 
ment of  the  views,  and  in  the  work  of  the  draughting  office,  as  well 
as  that  of  the  elementary  school-room,  no  further  use  is  made  of  the 
planes  ;  for  shop  and  elementary  drawings  are  not  arranged,  with 
reference  to  the  edges  of  the  planes,  as  are  drawings  in  which  pro- 
jection methods  are  used. 

33.  Teachers  who  prefer  to  consider  the  object  as  seen  from  one 
direction  and  turned  to  present  the  surfaces  to  be  represented,  must 
explain  the  arrangement  of  views  desired,  and  also  the  fact  that  a 
point  of  the  top  view  is  in  a  vertical  line  from  the  same  point  in  the 
front  view,  and  that  a  point  in  a  side  view  is  in  a  horizontal  line 
through  the  front  view  of  the  same  point.     Pupils  may  be  told  these 
facts  and  be  taught  to  make  their  drawings  in  accordance ;  but  the 
reasons  for  these  facts  are  most  easily  understood  when  they  are 
clearly  illustrated  by  use  of  the  slates. 

To  present  the  principles  as  explained  above  will  take  little 
longer  than  to  state  only  the  facts  regarding  the  desired  positions  of 
the  views  ;  when  the  principles  are  not  explained,  these  positions  are 
very  likely  to  be  forgotten. 

34.  After  the  pupils  understand  the  arrangement  and  nature  of 
the  views,  mention  of  the  slates,  or  planes  of  the  drawing,  is  unnec- 
essary, and  the  work  is  practically  the  drawing  of  different  views  of 
an  object,  which  is  so  turned  that  these  views  may  be  seen  and 
drawn  in  their  proper  relations  to  the  front  view. 

If  the  subject  is  presented  to  pupils  too  young  to  understand 
what  has  been  explained,  the  work  must  be  simply  dictation  or  copy- 
ing. Much  of  the  time  now  spent  on  working  drawings  in  the  lower 
grades  of  the  public  schools  is  wasted,  for  the  pupils  can  do  no  more 
than  copy  ;  instead  of  copying,  they  might  spend  their  time  profitably 
upon  free-hand  drawing,  and,  when  old  enough  to  reason,  take  up  the 
study  of  working  drawings,  and  thus  not  be  obliged  to  copy. 

35.  Making  Working  Drawings.  —  When  given  any  object  for 
which  drawings  are  to  be  made,  the  first  question  is,  what  position  of 
the  object  shall  be  represented,  and  the  next,  what  views  will  best 
show  its  construction  and  require  the  least  time  to  make. 


36  MECHANICAL   DRAW/A'i;. 

The  front  view  of  an  object  should  represent  its  front  surface, 
when  it  is  in  the  position  in  which  it  is  intended  to  be  used.  If  the 
object  has  no  portion  which  is  the  "front  surface,"  or  which  is  more 
important  than  the  others,  and  has  no  special  position,  its  position  is 
immaterial. 

Practical  drawings  give  no  more  views  than  are  needed  to  show 
all  the  construction  ;  but  while  studying  the  subject,  many  views  of 
simple  objects  may  be  made. 

36.  When  possible,  objects  for  study  should  be  placed  so  that 
one  principal  surface  will  appear  of  its  real  shape,  in  either  the  front, 
the  top,  or  the  side  view,  and  the  pupils  should  begin  with  this  view, 
or  with  the  one  concerning  which  they  know  the  most.     To  work 
advantageously  upon  subjects  at  all  complicated,  it  is  necessary  to 
begin  a  second  view  as  soon  as  the  first  view  represents  all  the  parts 
which  exhibit  their  real  shapes  in  it.     The  parts  which  show  their 
real  shapes  in  the  second  view  can  be  drawn  after  the  second  view 
has  given  all  that  the  first  view  represents,  and  then  the  parts  whose 
real  shapes  are  seen  in  the  second  view  may  be  projected  to  the  first 
view.     Circular  parts  should  always  be  represented  first  in  the  view 
in  which  they  appear  as  circles.     By  drawing  vertical  projecting  lines 
from  the  front  to  the  top  or  bottom  views,  or  horizontals  from  the 
front  to  the  side  views,  the  different  views  will  be  made  to  agree  with 
and  to  complete  one  another.     In  this  way  several  different  views  can, 
and  should  be  carried  along  together.     In  order  to  make  them  accu- 
rately and  rapidly,  the  dimensions  of  circles,  or  of  any  parts  which 
have  the  same  size  in  two  or  more  views,  should  be  taken  in  the  com- 
passes or  dividers  and  set  off  at  once  in  all  the  views.     This  method 
is  much  more  accurate  and  rapid  than  working  by  projecting  lines, 
which  often  are  not  drawn  quite  parallel ;  this  is  the  draughtsman's 
method,  and  though  it  may  not  be  possible  for  young  pupils  to  work 
in  this  way,  it  should  be  explained,  and  older  pupils  should  use  it. 

37.  Views  of  a  Circular  Plinth.  —  Fig.  8  is  a  perspective  of  a 

circular  plinth  of  which  working  drawings  are  to  be 
made. 

If  the  plinth  is  a  circular  box  it  will  be  placed 
horizontally ;  if  it  is  a  clock  it  will  be  placed  with  its 
FIG.  s.  circles  vertical.     In  general  the  object  may  occupy 


WORKING   DRA  WINGS. 


37 


FIG.  9. 


any  position,  but  it  is  better  to  have  it  placed  in 
the  position  in  which  it  is  intended  to  be  used. 
Suppose  the  circles  to  be  vertical  and  the  -object 
to  be  seen  from  -the  front,  so  that  the  front  surface 
appears  of  its  real  shape.  The  circle  f,  Fig.  9, 
will  then  be  the  front  view  of  the  object.  This 
circle  should  be  drawn  before  any  of  the  lines  of 
the  top  view  are  drawn. 

To  see  what  the  top  view  will  represent,  hold 
the  plinth  so  that  its  front  face  appears  a  circle,  as  in  F,  Fig.  9, 
and  then  turn  the  top  of  the  plinth  toward  the  eye  until  the  circles 
appear  as  straight  lines  ;  that  is,  as  nearly  straight  as  it  is  possible 
for  them  to  appear  when  held  in  the  hand.  In  this  position  the 
curved  surface  is  seen,  and  the  outline  of  the  object  is  very  nearly  a 
rectangle.  The  circles  cannot  appear  as  straight  lines  at  the  same 
time,  but  the  block  can  be  held  so  that  the  circles,  one  at  a  time, 
will  appear  straight ;  and,  since  the  circles  are  perpendicular  to  the 
top  plane,  this  is  the  appearance  which  both  must  present  in  the  top 
view,  7}  Fig.  9.  The  length  of  the  rectangle  of  the  top  view  must  be 
the  same  as  the  diameter  of  the  circle  F,  and  its  width  must  be  the 
same  as  the  thickness  of  the  plinth.  The  horizontal  lines  which 
represent  the  two  circles  must  then  be  as  far  apart  as  the  plinth  is 
thick,  and  the  short  vertical  lines  must,  if  extended,  be  tangent  to 
the  circle  of  the  front  view ;  they  should  be  placed  by  drawing  tan- 
gents (projecting  lines)  to  the  front  view,  or  by  setting  off,  with  the 
compasses,  the  required  distance  from  either  side  of  the  centre  line, 
as  explained  in  Art.  36. 

38.  Views  of  an  Hexagonal  Plinth.  —  Fig.  10  is  a  perspective 
of  an  hexagonal  plinth,  or  box,  of  which  working  drawings  are  to  be 
made  from  the  object. 

The  plinth  being  horizontal  when  in  use,  the  top 
hexagon  will  appear  its  real  shape  in  the  top  view. 
This  view  should  be  drawn  first,  the  plinth  being 
placed  so  that  if  any  vertical  faces  are  more  important 
than  others  they  may  be  seen  in  the  front  and  side 
views  of  the  object.  This  may  cause  the  hexagon  of  the  top  view  to 
have  a  long  diagonal  horizontal  or  vertical. 


FIG.  10. 


MECHANICAL    DRA  WING. 

Suppose  the  hexagon   T,  Fig.  1 1 ,  to  be  the  top  view  of  the  box, 

and  the  front  and  right  side  views 
to  be  required.  To  see  what  the 
front  view  will  represent,  the  box 
must  be  looked  at  horizontally 
when  placed  as  shown  in  the  top 
view  ;  or,  if  we  move  the  object  to 
present  the  different  views,  the  box 
i  J{  S.  should  be  held  so  that  its  hexagonal 
faces  are  vertical,  each  has  two 
edges  horizontal,  and  so  that  only  the  hexagonal  face,  originally  the 
top  of  the  object,  is  visible.  When  in  this  position  the  lower  horizon- 
tal edge  of  the  visible  hexagon  represents  the  face  of  the  plinth, 
originally  its  front  vertical  face.  The  object  is  now  to  be  revolved 
so  that  this  lower  edge  is  brought  upward  toward  the  eye  until  the 
hexagons  appear,  as  nearly  as  possible,  straight  lines.  This  position 
gives  the  appearance  which  the  front  view  must  represent;  in  this 
view  three  vertical  faces  are  visible  as  surfaces,  and  the  outline  of 
the  view  is  a  rectangle. 

Knowing  the  appearance  which  the  front  view  must  represent,  it 
may  be  drawn  in  its  proper  position  below  the  top  view.  The  verti- 
cal lines  of  the  front  view  are  equal  in  length  to  the  thickness  of  the 
plinth,  and  must  be  vertically  under  the  corners  of  the  hexagon  which 
in  the  top  view  represent  these  lines.  The  horizontal  lines  which 
represent  the  hexagons  are  thus  as  long  as  a  long  diagonal  of  the 
hexagon.  See  F,  Fig.  1 1 . 

To  obtain  the  right  side  view,  the  object  should  be  held  to  appear 
as  shown  in  the  front  view,  and  then  be  revolved  about  a  vertical 
axis,  the  right  vertical  edge  coming  toward  the  eye,  until  the  object 
is  seen  at  right  angles  to  the  direction  in  which  the  front  view  was 
visible.  In  this  position  the  two  visible  vertical  faces  will  appear  of 
equal  widths,  the  vertical  surface  originally  at  the  front  of  the  object 
will  appear  a  vertical  line  at  the  left  of  the  object,  and  the  vertical 
surface  originally  at  the  back,  a  vertical  line  at  the  right  of  the  object. 
The  distance  between  these  surfaces  is  the  length  of  a  short  diagonal 
of  the  hexagon.  These  facts  having  been  noted,  the  right  side  view 
may  be  drawn  in  its  proper  position  at  the  right  of  and  on  the  same 


WORKING   DRA  WINGS. 


39 


level  as  the  front  view,  its  width  being  the  distance  1-2  of  the  top 
view.  See  JR.S.,  Fig.  u. 

39.  Views  of  a  Plinth  and  Disc.  —  Fig.  12 
represents  an  equilateral  triangular  plinth  support- 
ing, by  means  of  two  wires,  a  vertical  circular  disc, 
which  appears  a  circle  in  the  front  view. 

To  represent  this  object  only  the  front  and  side 
views  are  necessary  ;  but  the  top  view  is  added. 

It  is  natural  to  draw  first  the  triangular  plinth, 
because  it  supports  the  disc.  The  disc  appears  a 
circle  in  the  front  view,  and  the  plinth  appears  a 
triangle  in  the  side  view,  so  it  is  evident  that  the 
two  views  should  be  carried  along  together  as  ex- 
plained in  Art.  36. 

The  side  view  S,  Fig.  13,  in  which  the  plinth  appears  a  triangle, 

should  first  be  drawn  ;  but  this 
view  cannot  be  completed  to  rep- 
resent the  circle  until  the  front 
view  of  the  circle  is  drawn  ;  for 
the  wires  ^  "  long,  which  support 
the  circle,  cause  its  lowest  point 
to  come  below  the  ends  of  these 
wires  an  unknown  distance. 

Having  completed  the  triangle 
of  the  side  view,  the  front  view  of 
the  plinth  should  be  drawn.  To 
discover  its  appearance,  hold  the 
plinth  in  the  position  represented 
by  the  side  view,  and  then  revolve 
the  object  on  a  vertical  axis  so 
that  the  right  hand  points  of  the 
plinth  move  toward  the  eye  until 
the  vertical  triangles  appear  as 
FlG-  I3>  nearly  as  possible  vertical  lines. 

One  sloping  face  is  then  visible  ;  the  horizontal  base  appears  a  hori- 
zontal line  whose  length  is  equal  to  the  thickness  of  the  plinth  ;  and 
the  triangles,  one  at  a  time,  appear  vertical  lines,  whose  length  is 


MECHANICAL   DKA  WING. 


equal  to  the  distance  1-2  of  S.  The  front  view  of  the  plinth  may 
now  be  drawn  at  the  right  of  and  on  the  same  level  with  the  side 
view.  In  this  view  the  wires  which  support  the  disc  are  seen  of  their 
real  lengths  ;  they  should  be  represented  when  the  front  view  of  the 
plinth  is  completed. 

To  obtain  the  circle  which  represents  the  disc,  describe  arcs,  whose 
radius  is  that  of  the  circle,  from  the  upper  ends  of  the  wires.  These 
arcs  intersect  at  j",  the  centre  of  the  circle,  from  which  the  circle 
may  be  described.  In  the  side  view  the  circle  appears  a  vertical 
line;  this  view  may  now  be  completed  by  limiting  this  line  by 
a  horizontal  projecting  line  tangent  to  the  circle  at  its  highest 
point. 

The  top  view  of  the  plinth  alone  might  have  been  given  before 
this,  but  as  there  are  no  parts  which  must  be  represented  first  in  the 
top  view,  the  time  of  drawing  it  is  not  important.  To  see  the  appear- 
ance that  the  top  view  represents,  hold  the  object  as  it  appears  in 
the  front  view,  and  then  revolve  the  top  toward  the  eye  until  the 
circular  disc  appears  a  horizontal  line.  In  this  position  the 
length  of  the  rectangle  which  represents  the  plinth  will  be  equal 
to  3-4,  or  the  length  of  the  base  of  the  object  as  seen  in  the 
side  view. 

The  appearance  having  been  discovered,  the  top  view  may  be 
placed  above  and  in  line  with  the  front  view,  by  means  of  verti- 
cal projecting  lines  through  the  points  of  the  front  view.  See  T, 
Fig.  13- 

40.  Views  of  a  Box  and  Pyramid.  —  Fig.  14  is  a  perspective 
of  a  tin  box  with  its  cover  opened  back 
and  resting  on  the  surface  that  supports 
the  box.  A  square  pyramid  is  centrally 
placed  on  the  top  of  the  box,  the  edges 
of  its  base  being  at  45°  to  those  of  the 
top  of  the  box.  The  view  which  shows 
the  front  of  the  box,  we  will  call  the  front 
view.  This  and  the  top  and  side  views 
are  required.  The  box  is  6"  long,  4" 
wide,  and  3"  deep.  The  base  of  the  pyra- 
FIG.  14.  rnid  is  3/^"  square  ;  its  axis  is  6". long. 


WORKING   DRA  WIXGS. 


FIG.  15. 


In  the  front  view  the  box  is  represented  by  a  horizontal  rectangle 
3"  high  and  6"  long.  This  may  be  drawn  first,  and  then  the  top  and 
the  end  views  of  the  box. 
The  top  view  of  the  box  is 
a  horizontal  rectangle  4" 
wide  and  6"  long  ;  the  side 
view  is  a  horizontal  rec- 
tangle 3"  high  and  4"  lon^, 
and  is  on  the  same  level 
as  the  front  view,  while 
the  top  view  is  directly 
over  the  front  view.  In 
the  side  view  the  cover  is 
seen  extending  obliquely 
from  the  top  of  the  box  to 
the  level  of  the  bottom. 
To  draw  the  side  view  of 
the  cover,  an  arc  of  a  cir- 
cle whose  radius  is  4" 
should  be  described  from  the  edge,  where  the  cover  swings  (point  3) 
to  intersect  the  line  of  the  table  in  2.  The  cover  must  be 
represented  by  a  line  drawn  from  2  to  3.  The  top  view  of  the  cover 
may  now  be  drawn  ;  its  width  in  this  view  is  the  distance  1-2  of  the 
side  view. 

In  the  top  view  the  square  base  of  the  pyramid  is  seen  of  its  real 
shape  ;  this  should  be  the  first  view  of  the  pyramid  drawn.  To  ob- 
tain the  square,  draw  the  diagonals  of  the  top  view  of  the  box;  these 
intersect  at  its  centre.  With  this  point  as  centre  describe  a  circle 
3^"  in  diameter.  Tangent  to  this  circle  and  at  45°  with  the  edges 
of  the  box,  draw  the  sides  of  the  square  which  represents  the  base  of 
the  pyramid.  After  this  is  drawn,  the  top  view  may  be  completed 
by  drawing  the  diagonals  of  the  square  to  represent  the  lateral  edges 
of  the  pyramid. 

In  the  front  and  side  views  the  base  of  the  pyramid  coincides 
with  the  top  of  the  box.  In  the  front  view  the  axis  of  the  pyramid 
is  a  vertical  line  just  under  the  centre  of  the  square,  —  the  point 
which  represents  the  axis  in  the  top  view. 


42  MECHANICAL   DRAWING. 

In  the  side  view  the  axis  is  a  vertical  line  midway  between  the 
front  and  back  of  the  box,  as  is  shown  by  the  top  view.  In  the  front 
view  the  width  of  the  base  is  given  by  projecting  from  a  and  b  in  the 
top  view  ;  and  in  the  side  view  the  width  of  the  base  must  be  equal 
to  c—  d  of  the  top  view.  From  c  and  d  in  the  side  view,  and  a  and  b  in 
the  front  view,  the  contour  lateral  edges  extend  to  the  top  of  the  axis. 
In  both  front  and  side  views,  the  nearest  lateral  edge  of  the  pyramid 
is  a  vertical  line  ;  this  line  represents  also  the  axis  of  the  pyramid. 

41.  It  will  be  difficult  to  hold  the  models  so  that  they  have  the 
proper  relations  and  present  to  the  eye  the  appearances  of  the  differ- 
ent views  of  the  complete  group.     The  box  and  the  pyramid  may  be 
held  separately  and  turned  to  give  the  appearances  of  the  different 
views  of  the  single  objects,  as  has  been  explained.     It  may  happen 
that  views  of  a  group  are  desired  when  the  relations  are  not  so  easy 
to  see  as  in  this  group  ; '  in  such  cases  it  will  be  much  better  to 
arrange  the  group  and  to  look  at  it  in  the  directions  of  the  different 
views  required. 

42.  The  making  of  simple  working  drawings  in  the  manner  ex- 
plained, requires  but  slight  knowledge  of  the  principles  of  projection. 
The  essential  points  are  that  the  front  and  top  views  of  the  same 
point  shall  be  in  the  same  vertical  line,  and  that  the  front  and  side 
views  of  any  point  shall  be  in  one  horizontal  line.    When  the  drawings 
are  made  from  the  object,  they  are  simply  representations  of  the  ap- 
pearances which  it  presents  when  seen  from  different  directions,  per- 
spective effects  of  vanishing  not  being  given.      Pupils  will  readily 
learn  to  look  at  an  object  from  different  directions  so  as  to  realize 
the  appearances  which  its    different   views  will  present;    they  will 
also  learn  to  revolve  the  object  to  present  these  different  appear- 
ances.   Both  methods  should  be  explained,  and  pupils  will  then  have 
little  difficulty  in  using  either. 

43.  The  simplest  way  to  illustrate  the  manner  in  which  an  object 
should  be  turned  to  present  the  appearances  which  the   different 
views  are  to  represent,  is  to  fold  about  the  front,  top,  bottom,  left, 
and  right  sides  of  a  cube,  squares  of  paper  which  form  the  develop- 
ment of  these  faces,  and  write  front,  top,  bottom,  left,  and  right 
sides  respectively  upon  these  different  surfaces.     If  the  cube  is  held 
so  that  the  front  view  is  visible,  and  the  other  marked  sides  occupy 


WORKING   DRAWINGS. 


43 


their  proper  positions,  and  if  the  squares .  upon  them  are  then  re- 
volved about  the  edges  of  the  front  face  into  the  plane  of  this  face, 
the  manner  in  which  the  cube  must  be  turned  to  present  the  different 
appearances  will  be  understood  at  once,  and  also  the  relations  of  the 
different  views  to  the  front  view.  This  experiment  should  be  made 
by  all  the  pupils. 

DIMENSIONING. 

44.  To  dimension  a  drawing  is  to  place  upon  it  all  measurements 
of  the  object  represented,  so  that  the  workman  may  construct  the  ob- 
ject from  the  given  measurements. 

In  order  to  be  easily  read  the  dimensions  should  be  so  placed  as 
not  to  crowd  or  interfere  with  each  other  or  with  the  lines  of  the 
drawing,  and  the  figures  must  be  neatly  made ;  poor  figures  will  spoil 
the  appearance  of  the  best  drawing. 

Dimensions  should  be  put  down  in  inches  and  fractions  of  inches 
up  to  two  feet,  thus :    16^". 

Distances  greater  than  two  feet  should  be  given  in  feet  and 
inches,  thus  :  3'  7|". 

Dimensions  should  be  placed  upon  dimension  lines  which,  by 
means  of  arrow-heads  at  each  end,  indicate  the  position  of  the 
dimension.  These  lines  should  not  be  continuous,  a  space  being 
left  to  receive  the  figures,  which  should  be  symmetrically  placed 
upon  the  line,  thus:  i  JkJ-"  ^J 

The  line  may  be  omitted    ^  ^ 

when  the  space  between  the  arrow-heads  is  short ;  and  when  there 
is  not  room  for  both  arrow-heads  and  dimension,  the  arrow-heads 
may  be  turned  in  the  direction  of  the  measurement  and      .      ~,, 
placed  outside  the  line,  thus :  I     'F  F 

When  there  is  not  room  even  for  the  dimension,  arrow-heads  may 
be  used  either  outside  or  inside  the  dimension  lines,  and  the 
dimension  placed  where  there  is  room  for  it,  thus  :  L  J 

Arrow-heads    and   figures    should    be    drawn    free-hand;      (_,. 
when  the  drawing  is  inked  they  should  be  drawn  with  a  com- 
mon  writing  pen    and    with  black  ink,   while   the  dimension 
lines  should  be  red.     The  line  separating  the  figures  indicating  the 
fraction  must  be  parallel  to  the  dimension  line. 


44 


MECHANICAL   DRAWING. 

Vertical  dimensions  should  be  placed  so  as  to  read  right- 
handed,  thus  : 

The  dimensions  may  be  placed  upon  the  drawing  when 
there  is  room  ;  but  when  the  space  is  small  it  is  better  to 
carry  the  dimension  outside  the  drawing  by  means  of  dot  and 
dash  lines,  thus : 


J 


I  The  space  between  the  different  views  is  often  the  best 

position  for  many  of  the  dimensions. 

When  an  object  is  divided  into  different  parts  and  the  lengths 
are  given  in  detail,  an  over-all  dimension  should  be  given. 

Dimensions  should  not  be  placed  upon  centre  lines. 

Distances  between  centres  of  all  parts,  such  as  rods,  bolts,  or 
any  evenly  spaced  parts,  should  be  given  ;  and  when  the  parts  are 
arranged  in  a  circle,  the  diameter  of  the  circle  passing  through  their 
centres  should  be  given. 

The  diameter  of  a  circle  and  the  radius  of  an  arc  should  be 
given.  The  centre  of  an  arc,  when  not  otherwise  shown,  should  be 
marked  by  a  small  circle  placed  about  it,  and  used  instead  of  an 
arrowhead.  The  dimension  line  should  begin  at  this  circle. 

Dimensions  should  be  clearly  given  in  some  one  view,  and  not 
repeated  in  other  views  ;  they  should  seldom  be  placed  between  a 
full  and  a  dotted  line,  or  between  dotted  lines  when  they  can  be 
placed  in  a  view  where  the  part  is  represented  by  full  lines. 

45.  Simple  objects,  circular  in  section,  are  often  shown  by  only 
one  view.     The  fact  that  they  are  circular  is  indicated  by  D.  or  Dia., 
understood  to  mean  diameter,  placed  after  the  dimension. 

46.  When  several  pieces  are  alike,  only  one  is  drawn,  and  the 
number  to  be  made  is  expressed  by  lettering. 

47.  When  parts  are  to  be  fitted  together,  it  is  customary  to  write 
whether  the  fit  is  to  be  "tight"  or  "loose." 

48.  Surfaces  which  are  to  be  finished  by  turning  or  planing  are 
often  indicated  by  placing  upon  a  line  drawn  parallel  to  their  outline, 
the  letter       which  means  finish. 


WORKING  DRAWINGS. 


45 


49.  Any  special  information  which  cannot  be  expressed  by  draw- 
ing is  always  expressed  by  lettering. 

The  name  of  the  object  represented,  the  scale  of  the  drawing,  its 
number  and  date  of  completion,  should  be  placed  upon  the  drawing. 

Simple  letters  should  be  used  for  all  this  work  ;  they  should  be 
small  and  not  more  prominent  than  the  drawing;  pupils  will  find 
block  letters  made  of  a  single  line  the  best  to  use. 

The  draughtsman  will  find  Soennecken's  system  of  round  writing 
neat  and  practical.1 

The  following  are  styles  of  lettering  suitable  for  use  : 

ROUND  WRITING. 


<w^34M<dffi\AM&vv\/v\, 

V  J  J      J 


0 


12£3  4561ZS  9  O 


COPYRIGHTED  BY  KEUFFBL  &  ESSBR,  1877. 

1  "This  is  not  a  -system  of  letteting,  but  a  scientifically  evolved  system  of 
writing  which  has  the  effect  of  lettering  without  requiring  the  same  amount  of 
skill  and  consumption  of  time." 


46 


MECHANICAL   DRA  WING. 


GEOMETRIC   LETTERS. 


C 


F 


H    I 


jklrnnapq_r 
stuvwxyz 


12345B7B9D 


GOTHIC   LETTERS. 


ABCDEFGHI 

JKLMNOPQR 

STUVWXYZ 

abcdefghi 


o    p     q 


j    k    I    m    n 

s     tu     vwx     y     z 

1234597890 


CHAPTER    IV. 


DEVELOPMENTS. 

50.  THE  development  of  any  object  gives  the  real  shapes  of  all 
the  surfaces  of  the  object.     It  is  obtained  by  unrolling  or  unfolding 
the  surface  and  placing  it  upon  a  single  plane  surface.    .Objects  are 
developable  or  non-developable,  according  as  their  surfaces  may  or 
may  not  be  laid  out  on  a  plane  surface. 

The  cube,  cone,  and  cylinder  are  types  of  developable  objects  ; 
the  sphere  is  a  type  of  an  undevelopable  object. 

51.  The  Cube.  —  The  development  of  the  cube,  Fig.  16,  may  be 

obtained  as  follows  :  Place  any  one  of  its  faces, 
as  A,  upon  paper,  and  with  a  sharp  pencil  trace 
its  outline.  Then  tip  the  cube  over,  revolving 
it  upon  the  edge  i  of  the  face  A,  until  a  second 
face,  B,  rests  on  the  paper  ;  then  trace  face  B*. 
In  the  same  way  trace  faces  C  and  Z>,  after 
revolving  upon  edges  2  and  3. 

The  development  of 
these  four  faces  is  a  rec- 
tangle whose  width  is 
equal  to  a  side  of  the 
cube,  and  whose  length 
is  equal  to  four  times  the 
side  of  the  cube.  After 
the  four  faces  A,  B,  C, 
and  D  have  been  placed 
upon  the  sheet  of  paper 
and  drawn,  revolve  the 
FlG-  17-  cube  so  as  to  bring  the 

faces  E  and  F\.Q  the  paper,  and  trace  them  ;  these  faces  should  be 
so  placed  that  one  edge  of  each  coincides  with  an  edge  of  the  face 
A,  />',  C,  or  D. 


FIG.   16. 


A 


F 
6 


J) 


48  MECHANICAL    DRAWING. 

The  real  shape  "of  each  face  is  thus  placed  upon  paper  in  what  is 
the  most  convenient  way,  supposing  the  cube  is  to  be  constructed  of 
tin,  paper,  or  any  other  thin  material  capable  of  being  bent ;  for  if 
the  material  is  cut  to  the  given  outline  and  then  bent  at  the  lines 
z,  2,  j,  4,  5,  and  6,  only  the  other  edges  of  the  object  require 
fastening  by  glue  or  solder.  See  Fig.  17. 

The  development  of  an  object  is  thus  a  pattern.  All  objects  of 
sheet  metal,  from  the  simplest  tin  dish  to  the  most  complicated  sheet- 
iron  work  in  pipes,  ventilators,  and  boats  are  obtained  by  means  of 
such  patterns. 

52.  When  the  form  developed  is  to  be  constructed  of  paper,  the 
outside  edges  of  the  development  should  be  provided  with  projecting 
pieces  called  laps,  by  which  the  different  parts  maybe  held  together. 
These  laps  are  shown   in  some  of  the  developments  given  in  the 
plates  of  this  book.     Pupils  making  the  drawings  should  construct 
the   forms   by   developing,  cutting,   folding,   and   gluing   the    parts 
together  ;  unless  the  objects  are  made  in  this  way,  many  of  the  pupils 
will  not  understand  the  subject  of  developments. 

The  paper  to  be  used  for  this  work  should  be  as  stiff  as  a  good 
quality  of  drawing  paper.  A  medium  weight  manilla  is  satisfactory. 
All  the  edges  to  be  folded  may  be  cut  partly  through  on  one  side  by 
a  knife,  to  cause  them  to  fold  sharply  and  neatly.  The  laps  should 
be  placed  inside  the  object  and  fastened  by  glue  or  thick  mucilage. 

Manual  and  mental  training  of  the  greatest  value  are  given  by  the 
making  of  objects  which  have  been  drawn  and  developed.  This 
work  has  the  advantage  of  being  equally  adapted  for  boys  and  girls, 
and  of  requiring  no  special  or  expensive  materials  or  tools. 

53.  When  these  experiments  have  been  made  and  understood, 
pupils  will  be  able  to  apply  the  principles   to  the  development  of 
solids,  represented  by  different  views,  without  using  the  object ;  or, 
if  the  object  is  used  to  obtain  the  views,  it  will  not  be  necessary  to 
place  it  upon  the  paper  and  trace  its  different  surfaces. 

This  method  of  tracing  is,  at  best,  an  inaccurate  one  ;  its  chief 
value  is  to  illustrate  principles  so  that  pupils  may  understand  that 
the  development  of  any  object  gives  the  actual  size  and  shape  of 
every  one  of  its  plane  surfaces,  and  the  actual  dimensions  of  all  of 
its  curved  surfaces  which  are  developable.  After  these  points  are 


DE  VELOPMRNTS. 


49 


understood,  pupils  will  be  able  to  make  accurate  developments  by 
working  from  the  different  views  of  an  object. 

54.  Prisms.     The   development  of  any  prism  will  be  obtained 
by  placing  its  different  faces  upon  the  paper,  and  tracing  their  real 
forms  upon  it.     The  order  in  which  these  surfaces  are  drawn  is  of 
little  consequence.     The  most  natural  way  is  to  revolve  the  object 
in  one  direction,  tracing  the  different  faces,  until  all  the  lateral  faces 
have  been  drawn.     The  bases  may  then  be  drawn  so  that  a  side  of 
each  coincides  with  an  equal  side  of  any  one  of  the  faces  already 
developed. 

55.  The  Cylinder.     The  development  of  the  curved  surface  of 

a  cylinder  will  be  obtained 
by  rolling  the  cylinder  along, 
and  tracing  as  it  moves, 
until  its  entire  curved  sur- 
face has  passed  over  the 
paper. 

A  right  cylinder  will  thus 
produce  a  rectangle  whose 
width  is  equal  to  the  length 
of  the  cylinder,  and  whose 
length  is  equal  to  the  cir- 
cumference of  a  base  of  the 
cylinder. 

In  exact  work  the  cir- 
cumference of  the  circular 
base  should  be  calculated,  but  in  practice  it  may  be  obtained  by 
dividing  the  circle  into  such  a  number  of  equal  parts  that  the  differ- 
ence in  the  distances  between  two  points  in  the  circle  measured  in  a 
straight  line  (the  chord)  and  measured  on  the  circle  (the  arc)  is  very 
slight. 

56.  By  using  the  dividers  and  spacing  the  circle  accurately,  the 
study  of  working  drawings  may  be  carried  on  without  the   introduc- 
tion of  mathematics  ;  if  the  points  in  the  circle  are  not  too  far  apart, 
this  method  is  accurate  enough  for  much  practical  work.     The  fact 
that   this   method   is    only  approximate   should  be  explained,   and 
when  the   pupils    are  older  and  have   had   sufficient  practice  with 


FIG.  18. 


MECHANICAL   DRAWING. 


instruments  suitable  for  accurate  work,  they  should  calculate  the  cir- 
cumference. This  cannot  be  done  before  the  pupils  enter  the  high 
school,  and  not  often  then,  for  the  instruments  generally  provided 
are  such  that  perfect  work  cannot  be  obtained  by  any  method. 

57.  In  Fig.  1 8  the  circle  is  divided  into  twelve  parts,  which  for 
most  work  in  the  public  schools,  will  be  found  better  than  a  larger 
number,  since  the  pupils  do  not  work  accurately  enough  to  warrant 
smaller  spaces  on  the  circle. 

If  a  good  pair  of  spring  dividers  could  be  used  and  the  entire  circle 
accurately  spaced,  twenty-four  parts  would  be  preferable  to  twelve  ; 
but  the  pupils  will  do  all  that  can  be  expected,  if  they  understand 
the  principles  and  succeed  in  making  models  which  are  neat  illustra- 
tions of  the  forms. 

The  bases  of  the  cylinder  are  circles.  In  the  development,  they 
may  be  drawn  tangent,  at  any  points,  to  the  sides  of  the  rectangle 
which  represent  the  development  of  the  edges  of  the  bases. 

58.  The  Cone.     The  curved  surface  of  the  right  cone,  Fig.  19, 
will  be  developed  by  rolling  the  cone  and  tracing  as  it  moves  until 
all  its  convex  surface  has  coincided  with  the  plane  of  the  paper. 
If  a  straight  line,    V—<j.,  drawn  on 

the  cone  from  the  vertex  to   the 

base,  is  placed  upon  the  paper,  and 

the    cone    then    rolled    until    this 

line  returns  to   the  paper,  all  the 

curved  surface  will  have  rolled  over 

the  paper.     All  the  lines  that  may 

be  drawn  from  the  vertex  to  the 

base  are  the  same  length.     As  the 

cone  rolls,  it  moves  about  its  vertex 

V,  which  remains  stationary.     One 

after  another    the    different    lines, 

or  elements,   of   its  surface  come 

to  the  paper,  and,  as  they  are  of  equal  length,  the  development  of 

the  circumference  of  the  base  is  an  arc  of  a  circle  whose  centre  is 

the  vertex  of  the  cone.     The  radius  of  this  arc  is  the  length  of  the 

straight  line  V~4,  from  the  vertex  to  the  base,  and  its  length  is  equal 

to  the  circumference  of  the  base  of  the  cone. 


FIG.  19. 


DE  VELOI'MENTS. 


To  determine  the  length  of  the  arc  bounding  the  development  of 
the  lateral  surface,  divide  the  base  of  the  cone  into  twelve  equal 
parts,  as  in  the  case  of  the  cylinder,  and  set  off,  upon  the  arc,  one 
of  these  spaces  twelve  times. 

The  development  of  the  base  of  the  cone  is  a  circle  equal  to  the 
base.  This  may  be  drawn  tangent  to  the  arc  at  any  desired  point. 

59.  Pyramids.     Fig.  20  represents  a  square  pyramid,  whose  sur- 
face may  be  developed  by  placing  any  edge,  as  i-  V,  upon  the  paper 

and  revolving  the  object  until 
its  four  triangular  faces  have 
been  traced  as  they  have  coin- 
cided with  the  paper.  The 
triangles  are  equal  and  isosce- 
les ;  their  bases  form  the  sides 
of  the  base  of  the  pyramid. 
The  lateral  edges  of  the  pyra- 
mid are  of  equal  length,  and 
in  the  development,  extend 
from  the  vertex  V  to  equidis- 
tant points  in  an  arc  whose 
radius  is  the  length  of  the 
lateral  edges.  The  distance 
1-2,  2-j,  3-4  and  4-1  in  this 

arc,  is  equal  to  the  length  of  a  side  of  the  base  of  the  pyramid. 

The  development  of  the  square  base   of   the  pyramid   may  be 

placed  so  that  one  side  of  the  square  coincides  with  a  base  of  any 

one  of  the  triangles  forming  the  lateral  surface  of  the  pyramid.     In 

this  way,  any  regular  pyramid  may  be  developed. 

The  distance  from  V  to  i  or  j  will  give  the  real  length  of  the 

lateral  edges  only  when  the  top  view  shows  that  these  edges  are 

parallel  to  the  front  plane. 

60.  The  Sphere.     The  solids  whose  surfaces  have  been  devel- 
oped illustrate   the   way   in   which  the   development    of    any   solid 
bounded  by  faces  or  by  surfaces  of  one  curvature  may  be  obtain'ed. 
The  sphere  is  bounded  by  a  surface  which  is  curved  in  every  direc- 
tion ;  this  surface  cannot  be  laid  out  upon  a  plane,  for  the  sphere 
touches  a  plane  in  one  point  only  and  as  it  rolls  describes  a  line 


FIG.  20. 


52 


MECHANICAL   DRA  WING. 


upon  the  plane.     The  surface  of  the  sphere  is  similar  to  those  of  . 
many  other  solids  which  are  curved  in  more  than  one  direction  and 
thus  cannot  be  developed. 

Though  the  surface  of  the  sphere  cannot  be  developed,  a  spheri- 
cal or  other  surface  of  double  curvature  may  be  covered  with  paper, 
metal,  or  other  thin  substance,  by  using  many  small  pieces,  each  of 
which  stretches  slightly  in  places  and  is  compressed  in  other  places, 
so  that  practically  they  cover  the  surface.  The  globe  is  thus 
covered  with  narrow  strips  of  paper  which  taper  to  a  point  in  oppo- 
site directions.  Each  of  these  strips  is  the  development  of  the 
curved  surface  of  one  of  the  equal  surfaces  of  a  solid  bounded  by 
edges  which  are  equal  circles  intersecting  each  other  at  two  opposite 
points,  and  by  curved  surfaces  which  are  straight  in  one  direction 
and  connect  these  edges.  These  curved  surfaces  come  within  the 
surface  of  the  sphere,  but  if  the  surfaces  are  narrow  they  are  within 
that  of  the  sphere  only  a  very  short  distance.  In  the  case  of  a  globe 
covered  in  this  way  the  surface  of  the  sphere  may  be  supposed  to  be 
intersected  at  equal  intervals  by  planes  passing  through  a  diameter. 

The  surface  of  the  sphere  may  be  intersected  by  parallel  planes 
perpendicular  to  a  diameter  of  the  sphere,  and  its  surface  covered  by 
the  developments  of  the  surfaces  included  between  the  cutting  planes. 
These  planes  intersect  the  surface  in  parallel  circles,  any  two  of 
which  may  be  supposed  to  be  the  bases  of  the  frustum  of  a  cone ;  if 
the  two  circles  are  near  each  other  the  curved  surface  of  the  frustum 
will  come  but  slightly  within  that  of  the  sphere  and  in  practice 
a. spherical  surface  may  be  covered  by  assuming  and  developing  the 
surfaces  of  many  frusta. 

To  obtain  these  developments  the  real  dimensions  of  the  assumed 
surfaces  must  be  determined  by  measuring  the  real  length  of  each 
line  by  which  the  real  shape  is  found  in  the  view  in  which  it  appears 
its  real  length,  or,  if  its  real  length  cannot  be  seen  in  any  view,  by 
finding  it  as  explained  elsewhere. 

Any  surface  which  is  curved  in  two  or  more  directions  may  be 
developed  by  assuming  and  developing  cylindrical  or  conical  sur- 
faces which  approximate  the  given  surface. 

It  will  often  be  necessary,  instead  of  proceeding  in  any  of  the 
ways  explained,  to  divide  a  surface,  by  means  of  lines  drawn  upon  it, 


DE  VEL  OPAfENTS. 


53 


so  that  triangular  surfaces,  which  together  form  a  series  of  plane  sur- 
faces which  approximate  the  given  curved  surface,  may  be  supposed 
to  pass  through  these  lines. 

In  practical  work,  sheets  of  metal  cut  to  form  the  development  of 
the  assumed  plane  surfaces  will  bend  and  stretch  to  form  the  curved 
surface.  Such  developments,  require  advanced  knowledge  and  are 
not  explained  in  this  book. 

61.  The  simplest  way  to  present  the  principles  of  this  subject 
is  to  trace  around  the  surface  of  an  object  which  is  placed  upon 
paper  as  explained  in  this  chapter.  ,. 

When  developments  are  made  from  drawings,  it  is  not  important 
how  the  surfaces  are  placed  as  long  as,  when  folded  together,  they 
will  form  the  object.  The  surfaces  may  be  placed  as  they  would  be 
arranged  by  tracing  around  the  object,  or  as  they  would  be  arranged 
by  tracing  the  surfaces,  one  after  the  other,  upon  a  transparent 
plane  placed  in  front  of  the  object.  This  method  is  in  harmony  with 
the  arrangement  of  views  adopted  in  this  book,  and  therefore  the 
developments  in  the  plates  are  arranged  in  this  way. 


CHAPTER    V. 
SHADOW    LINES. 

62.  SHADOW  lines  are  wider  than  the  regular  lines  of  the  draw- 
ing ;  they  have  the  same  effect  as  cast  shadows  and  relieve  the  pro- 
jecting parts  so  as  to  produce  an  effect  of  perspective,  even  in  pro- 
jection, which  is  without  perspective.  They  thus  make  drawings 
easier  to  read.  If  they  cannot  be  applied  so  as  to  produce  this  re- 
sult, they  should  not  be  drawn. 

Some  draughtsmen  suppose  the  light  to  come  from  behind  the 
left  shoulder,  in  the  direction  of  the  diagonal  of  a  cube  which  is 
so  placed  that  the  front  view  of  the  diagonal  is  a  line  at  45°  ex- 
tending downward  to  the  right,  and  the  top  view  of  the  diagonal  is  a 
line  extending  at  45°  upward  to  the  right.  The  shadow  lines  in  this 
case  will  generally  be  the  lower  and  right  hand  lines  of  the  front 
view,  and  the  upper  and  right  hand  lines  of  the  top  view. 

Suppose  a  cube  resting  upon  a  corner,  to  be  revolved  about  a 
vertical  axis.  As  the  cube  revolves  the  shape  of  its  cast  shadow  is 
continually  changing,  and,  since  this  cast  shadow  is  composed  of  the 
shadows  of  the  different  edges  which  separate  the  surfaces  in  light 
from  those  in  shadow,  it  is  evident  that  these  lines  are  continually 
changing.  This  being  the  case,  no  rule  or  conventional  method  will 
enable  one  to  apply  shadow  lines  to  any  given  position  of  the  cube, 
so  that  they. shall  represent  the  edges  which  actually  separate  light 
from  shadow.  To  find  these  edges  we  must  obtain  the  cast  shadow 
of  the  cube ;  for,  except  in  very  simple  cases,  it  is  impossible  to  say, 
without  finding  the  cast  shadows,  which  are  the  edges  that  separate 
the  light  from  the  shadow  surfaces.  This  is  a  very  complicated 
problem,  and  hence  shade-lining  the  edges  that  separate  light  from 
shadow  is  not  suited  to  the  requirements  of  practical  work.  Not 
only  this,  but  since  the  edges  separating  light  from  shadow  surfaces 
are  continually  changing  as  the  object  moves,  the  shadow  lines  when 
properly  placed  upon  the  drawing  cannot  convey,  at  a  glance,  infor- 
mation of  much  value. 


SHADOW  LINES. 


55 


63.  Shadow  lines  will  be  of  little  value  if  they  cannot  be  applied 
according  to  some  system  which  does  not  require  much  time  or 
knowledge  to  determine  them,  and  which  always  represents  the  same 
facts  of  form  in  the  same  way. 

Instead  of  determining  the  edges  which  really  cast  the  shadows, 
some  draughtsmen  place  shadow  lines  on  the  right  hand  and  on  the 
lower  lines  of  the  front  view,  and  on  the  upper  and  right  hand  lines 
of  the  top  view.  f  Conventionally  shaded  in  this  way,  the  result  is  very 
different  from  that  given  by  shading  the  actual  shadow  edges.  As 
clearness  is  the  only  point  sought,  it  is  not  wise  to  follow  the  assumed 
direction  of  the  light  even  to  this  extent  if  a  simpler  and  clearer 
method  can  be  used.  As  already  explained,  the  views  may  be  sup- 
posed to  be  made  upon  one  plane  by  turning  the  object  as  explained 
in  Arts.  32  and  43.  In  this  case  the  light  will  have  one  direction  in 
all  the  views  and  the  shadow  lines  will  come  in  the  same  position  in 
all  views,  and  thus  be  much  easier  to  determine  than  when  they  are 
the  upper  lines  of  the  top  view  and  the  lower  lines  of  the  front  view. 

This  being  the  case,  it  is  the  custom  of  most  draughtsmen  to 
treat  all  the  views  in  the  same  way,  as  if  the  light  came  from  behind 
and  at  the  left,  its  direction  being  45°  downward  to  the  right. 

Assuming  this  direction  in  all  the  views,  the  lower  and  right  hand 
outlines  of  all  projecting  parts  will  cast  shadows,  and  should  be 
shade-lined  as  illustrated  in  the  following  cases  : 


A  square  pipe  in  different  positions  is  represented  by  A,  B,  C, 
and  D.  The  direction  of  the  light  is  represented  by  the  arrow. 

64.  In  A,  the  shade  lines  are  the  lower  and  right  hand  lines  of 
the  outside,  and  the  opposite  lines  of  the  inside  of  the  pipe.  These 
lines  at  the  inside  are  the  upper  and  left  hand  lines  of  the  drawing, 
but  lower  and  right  hand  lines  of  the  top  and  left  sides  of  the  pipe  ; 
and  every  part  of  any  object  which  is  thus  situated  must  be  shade- 
lined  as  if  it  were  a  separate  object. 


MECHANICAL    DRAWING. 


65.  In  J3,  two  sides  of  the  pipe  are  parallel  to  the  light,  and  the 
lines  of  neither  are  shaded.     The  lower  line  of  the  upper  left  and  the 
lower  line  of  the  lower  right  side  of  the  pipe,  are  right  hand  lines 
and  are  therefore  shaded. 

66.  In  C,  1-2  makes  an  angle  less  than  45°  with  a  horizontal 
line,  and  is  therefore  called  a  lower  line  and  is  shaded,  as  is  also  the 
parallel  upper  line  on  the  inside.     Line  2-4.  and  the  parallel  upper 
line  on  the  inside  are  right  hand  lines  and  are  shaded. 

67.  In  Z>,  3-4  makes  an  angle  with  a  horizontal  line  greater  than 
45°;  it  is  called  a  right  hand  line  and  is  shaded,  as  is  also  the  lower 
parallel  line  on  the  inside  of  the  pipe.     Line  2-4  and  the  upper 
parallel  line  on  the  inside  are  lower  lines  and  are  shaded. 

68.  E  represents  a  cylindrical  pipe.     Here  the  out- 
side circle  from  /  to  2  (points  in  a  diameter  at  right 
angles  to  the  light)  is  shaded  below,  and  the  inside  circle 
is  shaded  above  points  j  and  4,  which  are  in  the  same 
diameter. 

69.  /''represents  a  cylinder  from  whose  end  projects  a 
smaller  cylinder.     Here  both  circles  are  shaded  below  the 
diameter  at  right  angles  to  the  light. 

70.  G  and  Zfare  parts  of  the  same  object ;  If  is  cylindrical,  G 

is  square,  and  has  its  sides  at  45°  to  the 
plane  of  the  drawing.  From  If  a.  square 
part  projects,  and  from  G  a  round  part. 
It  is  generally  customary  not  to  shade 
the  element  of  a  curved  surface,  and,  therefore,  if  this  is  understood, 
a  glance  at  this  one  view  would  indicate  that  the  part  H  is  not  square, 
and  has  no  sharp  edge  represented  by  its  lower  outline;  the  same 
is  true  regarding  the  projection  from  G,  and  the  first  impression 
produced  is  that  these  parts  are  round.  If  they  are  not  round,  this 
fact  will  be  shown  by  the  other  views.  The  shade  lines  on  the  lower 
part  of  G,  on  the  projection  from  If,  and  on  the  right  verticals  of  the 
right  hand  parts,  emphasize  the  fact  that  there  are  none  on  the  round 
parts.  The  right  hand  Jine  of  Zf  is  shaded  because  it  represents  an 
edge  or  a  plane  surface. 

Study  of  these  illustrations  will  show  how  much  drawings  are 
improved  in  effect,  and  how  much  easier  they  are  to  read,  when  the 


SHA  DO  W  LINES. 


57 


shade   lines  are  given  by  this  simple,  conventional  method,   which 
may  be  stated  as  follows  : 

The  lower  lines  and  the  right  hand  lines  of  all  objects  are  shaded 
when  these  lines  represent  edges  or  surfaces,  but  are  not  shaded  when 
they  represent  the  elements  of  curved  surfaces,   unless  there  are  plane 
surfaces  extending  back  from  these  elements. 

71.  In  addition  it  should  be  said  that  the  draughtsman  must  use 
discretion,  and  give  or  omit  the  shade  lines  as  will  produce  the  best 
effect.  The  student,  however,  should  shade  all  drawings  as  explained. 

The  lines  of  sections  should  be  shaded  according  to  the  above, 
just  as  if  the  object  shown  in  section,  or  in  elevation  behind  the 
section,  were  entire  and  complete  in  itself. 

Shadow  lines  should  be  about  twice  the  width  of  the  regular  line, 
and  should  come  inside  the  line,  so  as  not  to  change  the  dimensions 
of  the  parts. 

After  the  drawing  has  been  completed  in  light  working  lines,  the 
pupil  should  go  over  all  the  lines,  making  them  of  a  width  and 
strength  suitable  for  result  lines  ;  he  should  then  give  the  required 
width  to  the  shadow  lines  by  going  over  them  again.  The  draughts- 
man may  ink  a  drawing  by  going  over  all  lines  not  shade  lines,  with 
the  regular  result  line,  and  not  inking  the  shadow  lines  until  they  are 
put  in  at  once  of  their  proper  width. 


CHAPTER    VI. 


INKING. 

72.  PUPILS  of  the  public  schools,  in  any  except  advanced  high 
school  classes,  should  not  attempt  to  finish  draw- 
ings in  ink,  as  they  will  obtain  the  best  training 
and  best  results  from  the  use  of  the  pencil. 

To  ink  accurately,  it  is  imperative  that  an 
exact  pencil  drawing  should  first  be  made,  its 
lines  being  fine  and  clear,  and  the  centres  of 
all  arcs  being  carefully  placed  and  marked  by  a 
small  free-hand  circle  about  them. 

A  special  pen,  called  a  drawing  pen,  and  also 
special  ink,  are  required  to  ink  a  drawing.  The 
best  pen  for  inking  is  one  without  a  hinged  joint, 
having  its  outer  blade  more  curved  than  the 
inner  one. 

The  illustration  represents,  full  size,  the  lower 
portion  of  a  pen  suitable  for  students'  use. 

73.  India  Ink.  The  ink  to  be  used  may  be 
liquid  India  ink  which  is  sold  in  bottles,  or  ink 
prepared  as  it  is  needed,  by  grinding  the  solid 
stick  India  ink. 

The  liquid  ink  is  prepared  with  chemicals  which  cause  it  to  enter 
the  paper  so  that  its  lines  are  erased  with  much  more  difficulty  than 
those  made  from  stick  ink,  which  is  ground  as  it  is  needed.  Highly 
finished  ink-drawings  will  be  made  more  easily  if  the  ink  used  is  pre- 
pared by  grinding,  but  the  trouble  of  preparing  it  may  render  the 
use  of  the  liquid  ink  advisable. 

The  stick  ink  may  be  prepared  for  use  by  grinding  it  in  pure  water, 
in  an  ink  slab  which  may  have  either  of  the  sections  illustrated.  The 
stick  must  be  kept  in  motion  all  the  time,  slight  pres- 
sure being  applied,  until  the  ink  is  thick  enough  to 
give,  when  dry,  a  perfectly  black  and  solid  line.  After 
the  ink  is  prepared,  the  stick  should  be  carefully  wiped 
to  prevent  its  crumbling. 


INKING. 


59 


The  slab  must  be  kept  covered  to  prevent  evaporation  and  to 
keep  the  ink  free  from  dust.  If  the  ink  hardens  in  the  slab,  it  must 
be  washed  out.  Fresh  ink  must  be  prepared  every  few  days,  as  it 
spoils  quickly. 

74.  Inking  a  Drawing.  The  pen  may  be  filled  with  a  quill 
toothpick.  It  is  not  necessary  to  move  the  blades  to  fill  the  pen ; 
they  should  be  set  for  the  proper  line  and  remain  unchanged  until  all 
lines  of  that  width  are  inked.  When  filled,  the  outside  surfaces  of  the 
blades  should  be  wiped  with  a  cloth  or  chamois-skin  to  remove  any 
ink  upon  them.  If  any  remains  upon  that  of  the  inside  blade  it  will 
soon  find  its  way  to  the  straight  edge,  and  then  to  the  paper.  The 
pen  must  be  thoroughly  cleaned  after  using,  to  prevent  the  blades 
from  rusting. 

The  pen  should  be  held  with  the  inside  blade  against  the  straight 
edge  or  French  curve,  and  parallel  to  its  edge.  If  it  is  turned  so 
that  the  blades  are  at  an  angle,  the  ink  will  flow  to  the  straight  edge 
or  curve  and  blot  the  paper.  To  keep  the  pen  in  the  proper  posi- 
tion, the  forefinger  may  be  placed  upon  the  set  screw,  or  the  thumb 
may  be  placed  on  the  edges  at  one  side  and  the  fingers  on  the 
edges  at  the  opposite  side  of  the  blades. 

The  pen  should  be  held  with  slight  and  even  pressure  against  the 
straight  .edge  or  curve.  If  the  pressure  varies,  the  blades  will  spring 
and  the  width  of  the  line  will  change.  Both  blades  of  the  pen  should 
bear  equally  upon  the  paper.  A  ragged  line  results  when  only  one 
blade  touches  the  paper.  To  correct  this  defect  the  pen  must  be  in- 
clined in  the  direction  of  the  broken  side  of  the  line,  until  both 
blades  bear  equally  upon  the  paper. 

The  blades  should  be  of  such  length  that  both  will  bear  equally 
upon  the  paper  when  the  pen  is  inclined  slightly  so  as  to  bring  the 
inner  blade  near  the  straight  edge.  If  the  pen  does  not  make  a  good 
line  when  held  as  directed,  it  must  be  sharpened ;  but  this  cannot  be 
done  by  young  pupils. 

The  angle  of  the  pen  with  the  edge  of  the  straight  edge  must  not 
be  changed  while  drawing  any  line,  as  this  will  vary  the  distance  of 
the  point  from  the  straight  edge  and  produce  a  crooked  line. 

In  going  over  a  line  the  second  time,  the  pen  should  be  inclined 
and  moved  in  the  same  direction  as  when  the  line  was  first  drawn. 


60  MECHANICAL   DRAWING. 

If  a  pen  does  not  draw  a  smooth  line  without  pressure,  or  if  it 
cuts  or  scratches  the  paper,  it  should  be  sharpened.  It  will  require 
sharpening  often  if  it  is  used  frequently,  for  the  blades  are  quickly 
dulled  by  the  paper. 

When  inking  circles  or  arcs,  the  pen  point  must  be  so  inclined 
that  both  blades  bear  equally  on  the  paper. 

To  ink  very  small  circles  a  good  bow  pen  is  necessary,  which, 
together  with  hair-spring  dividers,  will  be  required  whenever  really 
fine  work  is  desired. 

All  circles  and  arcs  should  be  inked  first;  next  the  horizontal  lines, 
beginning  with  those  at  the  top  of  the  sheet  and  working  downward ; 
then  the  vertical  lines,  beginning  with  those  at  the  left  of  the  sheet 
and  working  toward  the  right.  All  parallel  lines,  when  not  hori- 
zontal, vertical,  or  at  any  of  the  angles  given  by  the  triangles,  should 
be  inked  at  one  time,  by  means  of  triangles  used  as  explained  in  Art.  6. 

If  the  lines  are  not  inked  in  order,  from  the  top  downward  and 
from  left  to  right,  they  will  be  blotted  by  placing  the  straight  edge 
upon  them  before  they  are  dry. 

To  obtain  perfectly  tangential  arcs,  the  tangent  point  must  be 
found  by  drawing  a  pencil  line  connecting  the  centres  from  which 
the  arcs  are  drawn  ;  to  obtain  perfectly  tangen- 
tial straight  lines  and  arcs,  the  tangent  points 
must  be  found  by  drawing  a  straight  line  from 
the  centre  of  each  arc,  perpendicular  to  each 
straight  line  to  which  it  is  tangent.  The 
draughtsman  may  work  without  these  aids,  but  they  are  required 
by  students. 

Hatching  lines  should  be  a  little  finer  than  outlines,  and  should 
not  be  placed  nearer  together  than  is  necessary  to  avoid  the  effect  of 
a  series  of  bars  or  wires,  which  will  be  given  when  they  are  too  far 
apart.  They  should  present  the  appearance  of  a  tint :  from  ten  to 
twenty  lines  per  inch  will  give  satisfactory  results. 

When  many  working  lines  radiate  from  a  point,  all  should  not  be 
inked  to  the  point,  as  this  would  form  a  blot ;  they  should  stop  at 
unequal  distances  from  the  point. 

In  inking  a  symmetrical  curve,  such  as  the  ellipse,  part  of  the 
curve  each  side  of  the  axis  should  be  struck  with  the  compasses  from 


INKING. 


61 


a  Centre  in  the  axis  of  the  curve.     This  should  be  done,  even  if  no 
more  than  \  or  \  of  an  inch  can  be  drawn  in  this  way. 

When  the  surface  of  the  paper  has  been  roughened  by  erasing,  it 
may  be  made  smooth  by  rubbing  it  with  the  clean,  polished,  rounded, 
ivory  handle  of  a  knife  or  other  article. 

75.  Different  materials  are  shown  in  inked  drawings  by  using 
different  colors  for  the  line  sectioning,  or  by  tinting  the  sections  with 
washes  of  different  colors.     Centre  lines  are  generally  inked  full  red ; 
cast  iron  is  sectioned  black;  wrought  iron  or  steel,  blue;  brass,  yel- 
low ;  other  materials  are  represented  by  other  conventional  methods. 

76.  Blue  Prints.     Shop    drawings    are    generally   blue    prints. 
These   are   really   photographs   printed   from   a   drawing   made    on 
tracing  cloth.     The  lines  are  white  on  a  blue  ground. 

When  a  drawing  is  finished  wholly  in  black  lines,  or  made  for  re- 
production by  the  blue  print  process,  different  materials  are  shown 
by  different  hatchings.  The  materials  commonly  used  are  often 
represented  as  follows: 


Cast  iron. 


Wrought  iron. 


Steel. 


Composition. 


Babbitt  metal. 


ISS^ 


mt 


Vulcanite. 


Wood. 


Leather. 


Brick. 


Stone. 


77.  Erasing  and  Cleaning.  The  student  should  make  erasures 
in  an  inked  drawing  wholly  by  the  use  of  pencil  or  ink  erasers,  for 
when  the  knife  is  used  for  this  purpose  the  surface  of  the  paper  is 
injured,  so  that  the  lines  cannot  be  inked  again  neatly.  The  pencil 


62  MECHANICAL   DRAWING. 

eraser  will,  if  used  for  several  minutes,  remove  ink  lines  without 
injuring  the  surface  of  the  best  drawing  papers.  The  ink  eraser 
removes  the  lines  more  quickly,  and  generally  gives  satisfactory 
results. 

When  the  inking  is  finished  the  whole  drawing  may  be  cleaned 
by  rubbing  it  with  bread,  which  is  not  greasy  or  so  fresh  as  to  stick 
to  the  paper.  If  the  paper  is  much  soiled  it  may  be  necessary  to  use 
an  eraser.  A  soft  pencil  eraser  should  be  used  and  great  care  taken 
that  the  ink  lines  are  not  lightened  and  broken  by  it. 

To  avoid  the  necessity  of  using  an  eraser  upon  a  finished  draw- 
ing, instruments  and  paper  must  be  kept  free  from  dust  and  dirt. 
The  triangles  and  T-square  should  be  cleaned  often,  by  rubbing  them 
vigorously  upon  rough  clean  paper. 

78.  To  Sharpen  the  Pen.  —  The  blades  of  the  pen  should  be 
curved  at  the  points,  and  elliptical  in  shape.  To  sharpen  the  pen, 
screw  the  blades  together  and  then  move  the  pen  back  and  forth 
upon  a  fine  oil-stone,  holding  it  in  the  position  it  should  have  when 
in  use,  but  moving  it  so  that  the  points  are  ground  to  the  same 
length,  and  to  an  elliptical  form.  When  this  form  has  been  secured, 
draw  a  folded  piece  of  the  finest  emery  paper  two  or  three  times  be- 
tween the  blades,  which  are  pressed  together  by  the  screw.  This 
will  remove  any  roughness  from  the  inner  surfaces  of  the  blades ; 
these  surfaces  should  not  be  ground  upon  the  oil-stone. 

When  the  blades  are  ground  to  the  proper  shape,  they  must  be 
placed  flat  upon  the  stone  and  ground  as  thin  as  possible  without 
giving  them  a  cutting  edge.  To  do  this,  the  pen  should  be  moved 
back  and  forth  and  slightly  revolved  at  the  same  time.  Both  blades 
must  be  made  of  equal  thickness.  If  either  blade  is  ground  too 
thin,  it  will  cut  the  paper  as  would  a  knife,  and  the  process  must  be 
repeated  from  the  beginning.  In  order  to  see  the  condition  of  the 
blades,  they  should  be  slightly  separated  while  being  brought  to  the 
proper  thickness. 

79.    Stretching  Paper When  paper  is  to  be  stretched  it  should 

be  dampened  and  then  immediately  secured  to  the  drawing  board 
by  mucilage.  It  is  not  necessary  to  strain  the  paper  so  that  it  is  flat 
while  it  is  wet ;  it  is  better  not  to  do  this,  for,  if  this  is  done,  the. 
tension  created  in  drying  may  cause  the  paper  to  tear  if  the  board 


INKING.  63 

is  dropped,  and  when  the  paper  is  cut  from  the  board,  it  may  shrink 
enough  to  change  the  dimensions  of  the  drawing  a  sixteenth  or,  if 
the  drawing  is  large,  even  an  eighth  of  an  inch. 

80.    The  following  are  the  steps  involved  in  stretching  paper  : 

1.  Turn  over  and  press  down  for  about  ^"  the  edges  of  the 

paper,  all  the  way  around  the  sheet. 

2.  With  a  clean  sponge  and  water,  dampen  all  the  upper  surface 

of  the  paper,  including  that  of  the  folded  edges.  Allow 
the  sheet  to  stand,  and  moisten  again  if  necessary,  until 
the  paper  has  become  dampened  and  swollen  throughout. 
Time  may  be  saved  by  moistening  both  sides  of  the  paper. 
When  this  is  done  the  side  of  the  folded  edge  which  is  to 
receive  the  mucilage  should  be  dried  with  a  blotter  or 
cloth.  The  paper  should  not  be  rubbed  with  the  sponge, 
as  this  will  roughen  and  destroy  the  surface. 

3.  Apply  to  the  folded  edge  a  thick  mucilage  made  by  dissolving 

cheap  gum  arabic  in  cold  water. 

4.  Turn  the  edges  over  and  press  them  upon  the  board,  begin- 

ning in  the  centre  of  each  side  and  working  toward  the 
corners. 

5.  Press  the  edges  upon  the  board  and  rub  their  surfaces  until 

they  hold  firmly. 

6.  See  that  the  mucilage  holds  firmly  before  leaving  the  paper  to 

dry,  as  the  strain  will  pull  the  paper  from  the  board  if  the 
mucilage  does  not  set  while  the  paper  is  damp. 

7.  The  board   should   be  left  in  a  horizontal   position  with  no 

water  upon  the  surface  of  the  paper ;  it  should  never  be 
placed  near  a  radiator  or  in  the  sun. 

8.  The  edges  may  be  quickly  set  by  rubbing  them  with  a  piece 

of  polished  hot  metal. 

Mucilage  is  better  than  glue  for  stretching  paper,  as  it  does 
not  set  too  quickly,  and  the  edges  of  the  paper  may  be 
easily  removed  from  the  board  when  the  sheet  is  cut  off. 
To  remove  these  edges,  cover  them  with  water ;  after  a  few 
minutes  the  paper  will  -absorb  the  water  and  they  can 
readily  be  removed. 


CHAPTER  VII. 

MACHINE   SKETCHING  AND   DRAWING. 

81.  PROJECTION  forms  the  basis  of  practical  working  drawings  ; 
in  order  to  make  technical  drawings  one  who  understands  projection 
has  simply  to  become  familiar  with  the  different  conventional  repre- 
sentations and  methods  which  cause  practical  drawings  to  differ 
from  complete  projections  of  the  objects. 

The  draughtsman  finds  it  necessary  to  make  drawings  of  machines 
already  made,  as  well  as  of  those  which  he  designs.  To  do  this  he 
makes  free-hand  sketches  of  the  machine,  measures  all  its  parts,  and 
places  the  dimensions  upon  the  sketches,  never  making  more  views 
than  are  necessary  to  show  the  construction  of  the  object  repre- 
sented. He  then  returns  to  the  office  and  makes,  from  his  sketches, 
finished  drawings  to  scale. 

This  book  is  intended  to  explain  projection  principles  and  also 
the  making  and  arrangement  of  practical  working  drawings,  and 
illustrates  drawings  of  both  kinds.  The  objects  used  by  pupils  for 
study  must  be  simple  and  easy  to  obtain.  In  order  that  pupils 
may  understand  how  to  represent  more  difficult  subjects,  it  is  neces- 
sary to  give  more  views  than  would  be  required  to  make  the  objects. 
This  should  be  remembered  when  the  plates  are  studied. 

Projecting,  or  guiding,  lines  as  they  are  sometimes  called,  are 
not  given  in  practical  drawings,  and  are  given  in  this  book  in  the 
drawings  of  machine  details  only  when  they  are  necessary  to  show 
how  projection  principles  are  applied  to  special  points. 

In  order  to  make  complete  working  drawings  of  the  drawings 
previously  made  and  explained,  it  will  only  be  necessary  to  place 
dimensions  upon  them  and  to  indicate  the  material  to  be  used  and 
the  amount  of  finish  to  be  given  the  different  parts. 

In  the  plates  dimensions  or  dimension  lines  have  been  placed 
upon  a  sufficient  number  of  drawings  to  illustrate  the  way  in  which 
dimensions  should  be  placed. 


MACHINE   SKETCHING  AND  DRAWING.  65 

82.  Sections  are  of  great  value  to  the  machine  draughtsman,  as 
they  enable  him  to  dispense  with  many  complicated  views.     They 
are  usually  taken  horizontally  or  vertically,  but  may  be  taken  in  any 
direction  ;  they  are  often  taken  at  right  angles  to  parts  whose  real 
shapes  are  not  shown  by  views  of  the  outside  of  the  object.     The 
part  of  the  object  behind  the  cutting  plane  is  generally  shown  in 
addition  to  the  surface  cut  by  the  plane  ;  but  the  surface  cut  by  the 
plane  is  often  all  that  is  necessary  to  give  the  desired  information, 
and   is   all  that  is  given  in   many  drawings  representing  sections. 
The    section    of   the    arm    on    line  AB  in   Fig.   117  illustrates  this 
point.     Such  a  section  may  be  placed  as  illustrated.     It  is  often 
found  about  AB  and  on  the  part  sectioned ;  in  this  case  the  part 
is  often  represented  as  broken,  thus  leaving  a  space  for  the  section. 

83.  When  an  object  is  too  long  to  be  shown  of  its  entire  length, 
and  is  of  one  shape  throughout  or  for  any  great  distance,  the  central 
part  may  be  considered  removed,   each  end  being  represented  as 
broken,  and  the  entire  length  shown  by  the  arrow-heads  and  figures. 

The  positions  at  which  sections  are  taken  are  usually  indicated  by 
dot-and-dash  lines. 

84.  The  surface  of  the  material  cut  by  the  cutting  plane  is  cross- 
hatched  with  lines  which  are  generally  drawn  at  45°,  and  in  the  same 
direction  on  the  same  piece,  wherever  it   may  extend  or  however 
much  it  may  be  cut  up  or  intersected  by  other  pieces.     The  distance 
between  the  lines  must  be  the  same  throughout  all  the  surface  of  any 
one  piece. 

When  different  pieces  are  cut  by  the  plane,  they  should  be 
hatched  in  opposite  directions,  especially  if  there  is  no  space 
between  them.  When  three  or  more  pieces  are  cut  and  come 
together,  they  can  be  distinguished  by  a  difference  in  the  spaces 
between  the  hatching  lines. 

85.  When  objects  such  as  bolts,  rods,  or  other  solid  parts  lie  in 
the  plane  in  which  a  section  is  taken,  they  should  be  shown  in  eleva- 
tion ;  for  time  is  required  to  hatch  the  parts,  and  nothing  is  gained 
by  doing  this,  since  they  are  solid.     Fig.  125  illustrates  this  point. 

86.  It  is  not  necessary  that  the  section  be  taken  on  one  straight 
line ;  it  may  be  taken  so  as  to  produce  the  clearest  drawing.     Thus 
in  Fig.  119  a  vertical  section  through  the  upper  part  of  the  caster 


66  MECHANICAL   DRAWING. 

would  cut  the  frame  so  as  to  show  part  in  elevation  and  part  in 
section.  It  is  much  clearer  to  suppose  the  cutting  plane  to  pass 
obliquely  through  the  centre  of  the  frame,  and  to  suppose  the  wheel 
to  be  cut  by  a  vertical  plane,  than  to  adhere  to  the  facts  of  projection 
given  by  the  use  of  any  one  plane. 

87.  If  a  wheel  having  arms  is  situated  with  one  or  two  arms  in 
the  plane  of  the  section,  neither  arm  should  be  sectioned,  for  the 
sectional  view  would  then  not  differ  from   that  of  a  solid  wheel. 
Thus  in  Fig.  116  a  horizontal  section  of  the  wheel  would  cut  two 
arms,  but  should  be  represented  the  same  as  the  vertical  section. 

88.  The  aim  of  the  draughtsman  is  to  convey  a  clear  idea  of  the 
object  to  be  constructed,  and,  as  certain  parts  and  details  which  are 
common  to  various  classes  of  work  are  generally  perfectly  known  and 
always  recognized  from  even  one  view,  he  often  shows  a  part  (for 
instance  a  set  screw)  in  position  in  one  view,  and  omits  it  in  other 
views  where  simply  its  position  is  shown.     Instead  of  perfect  projec- 
tions he  is  content  to  make  conventional  drawings  ;  he  is  generally 
considered  free  to  use  his  judgment,  both  as  to  the  views  which  shall 
be  made  and  as  to  the  essentials  of  these  different  views  ;  he  never 
makes  more  views  of  an  object  than  are  needed  to  show  its  construc- 
tion. 

89.  It  is  necessary  to  make  drawings  which  show  all  parts  of  an 
object  in  their  proper  positions.     These  views  are  called  assembly 
drawings;    upon  them    are    placed    important   over-all    dimensions, 
distances  between  centres,  etc.     For  the  workmen,  detail  drawings 
are  prepared  which  give  as  many  views  as  are  necessary  to  show 
fully  all  the  different  parts  which  make  up  the  complete  machine. 

The  assembly  drawing  shows  these  details  in  the  positions  they 
occupy  when  in  use,  and  many  of  them,  owing  to  the  positions  they 
occupy,  may  not  be  clearly  shown.  The  detail  drawings  arrange 
these  parts,  without  regard  to  the  position  which  any  part  occupies 
in  the  machine,  so  that  their  forms  are  clearest  shown  by  the  fewest 
views. 

Other  points  are  explained  on  the  pages  opposite  the  plates, 
Chap.  XII. 


CHAPTER   VIII. 

ORTHOGRAPHIC    PROJECTION. 

90.  ORTHOGRAPHIC  PROJECTION  is  the  art  of  representing  an 
object  by  means  of  projections  or  views  made  upon  different  planes, 
at  right  angles  to  each  other,  by  the  use  of  projecting  lines  perpen- 
dicular to  the  different  planes  and  passing  through  all  the  points  of 
the  object. 

The  work  of  the  draughting  office  often  requires  no  thought  of 
the  planes  of  projection ;  hence  some  teachers  claim  that  reference 
to  these  planes  is  unnecessary  ;  they  assert  that  pupils  should  not  be 
asked  to  consider  problems  of  projection  which  are  difficult  for  most 
of  them  to  understand. 

It  is  easier  to  remember  rules  than  to  understand  and  apply  prin- 
ciples. Working  drawings  may  be  made  by  rule  with  far  less  effort 
on  the  part  of  the  teacher  and  pupil  than  when  they  are  made  as  the 
result  of  knowledge  of  projection  principles  ;  but  this  does  not  prove 
the  principles  useless  or  unnecessary. 

To  draw  from  the  object  is  very  simple,  and  is  all  that  can  be  ex- 
pected from  grammar  school  pupils;  sometimes  little  more  can  be 
done  in  high,  evening  drawing,  and  in  elementary  technical  schools; 
but  in  the  high  and  elementary  technical  schools,  there  may  often  be 
found  students  able  to  reason  and  to  understand  the  principles  of 
projection  well  enough  to  draw  the  views  of  a  simple  object  from  a 
written  description  of  the  object  and  its  position.  Such  students 
should  have  the  benefit  of  the  training  given  by  a  course  in  projec- 
tion. Many  pupils  unable  to  understand  projection  sufficiently  to 
draw  even  simple  objects  from  written  statements,  will  nevertheless 
be  interested  and  benefited  by  a  talk  upon  its  principles  ;  and 
teachers  should  give  instruction  in  the  principles  of  projection  when- 
ever pupils  are  able  to  profit  by  such  instruction. 

Any  draughtsman  ought  to  be  able  to -draw  without  the  object 
before  him  all  the  time,  or  without  being  given  in  whole,  or  in  part, 


68  MECHANICAL   DRAWING. 

one  or  two  views  of  the  object.  Yet  the  instruction  given  by  many 
teachers,  while  it  enables  pupils  to  draw  from  the  object,  to  complete 
unfinished  views,  and  to  add  other  views  when  some  are  given,  does 
not  give  the  capacity  for  representing  objects  conceived  in  the  mind 
in  definite  positions. 

91.  A  common  method  of  instruction  is  for  teachers  to  place  upon 
the  blackboard  two  views  of  an  object  and  ask  the  pupils  to  draw  the 
third ;  or  to  give  the  axis  or  a  line  or  two  of  drawings  which  are  to 
be  completed  by  the  pupils.  This  method  causes  the  students  to  re- 
member the  problem,  or  a  similar  one,  and  the  steps  involved  in  its 
working.  There  are  few  who  with  this  assistance  are  unable  to  make 
the  required  drawings. 

If  the  representation  of  constructed  objects  was  the  only  work  of 
the  draughtsman  these  methods  would  always  be  as  satisfactory  as 
they  are  in  the  simple  work  of  object  sketching  and  draughting, 
whether  from  the  object  constructed  or  imagined  ;  but  the  draughts- 
man often  has  problems  which  require  objects  to  be  placed  in  certain 
definite  relations  to  each  other.  In  drawing,  he  must  refer  to  some- 
thing, and  whatever  this  is  called,  it  is  the  plane  of  the  drawing  or  a 
parallel  plane.  If  he  is  unable  to  draw  a  line  which  makes  definite 
angles  with  the  ground  and  with  the  vertical  plane  of  the  drawing, 
he  cannot  represent  an  object  as  simple  as  a  cube  or  cylinder,  which 
is  to  have  some  definite  position. 

Whatever  may  be  done  with  the  most  elementary  work,  the 
advanced  student  should  be  able  to  draw  simple  geometric  solids 
from  a  description  of  the  solid  and  its  position.  To  do  this  the 
planes  must  be  thought  of ;  and  teachers  who  are  proclaiming 
against  the  use  of  projection  methods  will  do  great  harm  if  they  con- 
fine students  old  enough  to  understand  and  profit  by  these  methods, 
to  work  from  the  object,  and  they  must  be  thus  confined  when  pro- 
jection methods  are  not  used. 

Models  and  objects  are  necessary  in  the  first  study  of  work- 
ing drawings  or  of  projection,  but  they  should  be  dispensed  with  as 
soon  as  possible. 

Pupils  should  learn  to  see  the  object  mentally  when  its  dimen- 
sions and  relations  to  the  planes  are  given,  and  to  see  also  the  pro- 
jections of  the  object  upon  these  planes.  They  will  then  be  able  to 


ORTHOGRAPHIC  PROJECTION.  69 

draw  without  reference  to  models.  Only  in  this  way  is  it  possible  to 
understand  the  subject  of  projection;  and  it  should  be  realized  that 
those  who  are  protesting  against  such  study  cannot  produce  a  sub- 
stitute method  which  will  accomplish  the  same  results. 

92.  When  pupils  are  taught  to  make  working  drawings  solely  by 
observation  of  the  object,  it  is  impossible  for  them  to  work  in  any 
other  way  ;  they  cannot  draw  even  two  views  of  the  simplest  form, 
such  as  a  cone,  whose  axis  makes  an  angle  of  30°  with  the  ground 
and  45°  with  the  front  vertical  plane. 

In  giving  an  examination  in  the  subject  it  is  impossible,  unless 
reference  is  made  to  the  planes  of  projection,  to  call  for  two  views  of 
any  line  or  object  inclined  to  both  planes,  without  giving  lines  or 
views  which  tell  the  pupils  what  they  are  to  do,  so  that  the  work 
becomes  memory  instead  of  reasoning. 

The  objection  to  reference  to  the  planes  of  projection  is  absurd 
when  coming  from  those  teachers  who  give  to  their  classes  difficult 
problems  of  intersections  and  oblique  projections,  which  cannot  be 
solved  except  by  the  use  of  cutting  and  other  auxiliary  planes,  whose 
relations  are  much  more  difficult  to  understand  than  those  of  the 
planes  of  projection.  Though  not  named,  the  planes  of  projection 
must  still  be  practically  used ;  hence,  they  should  be  explained  and 
the  study  carried  on  by  projection  methods  when  pupils  attempt 
these  problems. 

93.  Teachers    of  drawing  in  any  school    above    the  grammar, 
ought  to  understand  projection   and    descriptive  geometry,  even   if 
these  subjects  are  not  explained  to  their  classes  ;  they  will  then  be 
able  to   answer  questions  relating   to    the    representation  of  form, 
abstract  or   concrete,  even  if   they  have    not   studied  the   various 
methods  in  which  the  principles  are  applied  to  the  practical  work  of 
the  mechanic. 

It  is  true  that  knowledge  of  descriptive  geometry  will  not  enable 
one  to  represent  details  of  construction  in  the  conventional  ways 
peculiar  to  the  architect  or  the  machine  draughtsman  ;  but  this 
knowledge  is  the  foundation  for  all  drawings,  and  conventionalities 
are  quickly  understood  and  applied  by  one  who  is  able  to  represent 
correctly  any  form  in  any  desired  position. 

94.  The  principles  of  projection  which  have  been  explained  in 


yo 


MECHANICAL   DRAWING. 


connection  with  the  study  of  working  drawings  cover  the  simple 
work  of  the  draughting  office,  and  are  all  that  the  teacher  in  the 
grammar  grades  requires  ;  but  teachers  of  drawing,  advanced  stu- 
dents in  the  high,  and  students  in  technical  schools,  should  carry  the 
study  farther.  For  their  use  the  following  drawings  and  explana- 
tions are  given,  and  will  be  all  that  are  required  for  knowledge  of 
the  simple  principles  of  orthographic  projection.  A  thorough  under- 
standing of  the  subject  requires  a  course  in  descriptive  geometry. 

95.  This  book  is  intended  for  teachers  of  public  and  elementary 
schools,  in  which  the  subject  of  working  drawings  is  more  important 
than  projection.     The  best  arrangement  of  views  for  working  draw- 
ings is  explained  in   Chapter  III.      This   arrangement  is  different 
from  that  given  by  the  planes  generally  used  in  the  study  of  pro- 
jection. 

The  drawings  of  the  plates  of  this  book  are  arranged  as  working 
drawings  should  be  arranged,  and,  in  order  that  there  may  be  no 
confusion,  the  following  notes  on  projection  make  use  of  planes  of 
projection  which  produce  the  arrangement  of  views  chosen  for  use 
in  the  study  of  working  drawings. 

Some  of  the  objects  represented  on  the  plates,  with  their  rela- 
tions to  assumed  planes  of  projection,  are  described  at  the  end 
of  this  chapter.  These  planes  are  not  represented  in  the  drawings, 
and  those  who  study  the  following  notes  may  test  their  knowledge  of 
the  subject  by  using  the  statements  as  test  questions,  and  making 
the  projections  required  to  represent  the  objects  described. 

96.  Projection   Principles.  —  The  front,  top,    side,  back,    and 
bottom  planes  of  projection  are  represented  in  Fig.  21. 

The  drawings  or  tracings  made  upon  these  planes  are  the  differ- 
ent projections,  or  views,  of  the  object.  The  perpendiculars  to  these 
planes,  by  means  of  which  the  views  are  obtained,  are  the  projecting 
lines.  The  projections  will  be  called  views  throughout  this  chapter. 

In  all  the  drawings  a  point  will  be  designated  by  a  letter  or  fig- 
ure ;  its  projections  by  the  same  letter  or  figure  with  that  letter  as 
exponent  which  represents  the  plane  upon  which  the  view  is  found. 
Thus  T  means  the  top,  fihe  front,  R  the  right  side,  L  the  left  side, 
B  the  back,  and  G  the  bottom  view.  To  avoid  confusion  the  expo- 
nents are  often  placed  upon  only  part  of  the  letters ;  but  in  any  view 


ORTHOGRAPHIC  PROJECTION. 


where  an  exponent  is  found  upon  any  letter,  the   same   exponent 
should  be  understood  when  reading  the  other  letters. 

97.  Axes  of  Projection.      Fig.  22  represents  the  top,  bottom, 
and  side  planes  when  they  have  been  revolved  to  coincide  with  the 
plane  of  the  front  vertical  plane. 

The  lines  in  which  the  planes  of  projection  intersect  are  called 
axes  of  projection.  The  different  axes  are  named  in  Figs.  21  and  22. 

When  the  planes  are  in  the  positions  shown  in  Fig.  21,  and  the 
surface  of  any  one  plane  is  seen,  the  other  planes  are  seen  edgewise 
and  are  represented  by  the  axes  of  projection.  Thus,  when  the  front 
plane  is  seen,  the  front  horizontal  axis  at  the  top  represents  the  top 
horizontal  plane ;  the  front  horizontal  axis  at  the  bottom  represents 
the  bottom  plane,  and  the  right  and  left  side  planes  are  represented 
by  the  right  and  left  vertical  axes. 

In  the  first  study  of  projection  the  front  vertical  plane,  the  top 
horizontal  plane,  and  the  two  side  vertical  planes  are  the  only  planes 
of  projection  which  are  generally  required.  Their  axes  of  projection 
will  be  designated  by  the  following  abbreviations :  F.  If.  Axis;  J?. 
H.  Axis;  L.  H.  Axis ;  R.  V.  Axis  and  L.  V.  Axis.  The  lower  hori- 
zontal plane  not  being  used,  the  terms  are  understood  to  refer  to  the 
upper  axes. 

98.  Views  of  a  Point.     Fig.  21  represents  a  point,  A,  situated 
between  the  planes  of  projection.     A  projecting  line  from  A  to  the 
front  plane  gives  AF,  which  is  the  front 

view  of  point  A  ;  a  projecting  line  from  A 

to  the  left  side  plane  gives  AL,  which  is 

the  left  side  view  of  A;  a  projecting  line 

from  A  to  the  top  plane  gives  AT,  the  top 

view  of  A;   projecting  lines  to  the  right 

side  plane,  to  the  back  plane,  and  to  the 

lower  plane  will  give  the  views  of  A  upon 

these  planes.     These  views  may  be  desig-  FlG 

nated  by  AK,  AB,  and  AG  respectively. 

99.  The   projecting   lines   from  A  to   the  vertical   planes   are 
horizontal  and  in  a  horizontal  plane  passing  through  A.     This  plane 
intersects  the  different  vertical  planes  in  four  horizontal  lines,  which 
are  at  equal  distances  from  the  top  plane,  and  form  a  rectangle. 


MECHANICAL    DKA  WING. 


When  the  planes  of  projection  are  revolved  into  the  plane  of  the 
front  plane,  these  horizontal  lines  form  one  straight  line  which 
passes  through  AF  and  contains  the  different  vertical  views  of  A. 
The  projecting  lines  from  A  to  the  front,  back,  top,  and  bottom 
planes  are  in  a  vertical  plane  which  is  perpendicular  to  the  front 
plane  and  parallel  to  the  side  planes.  This  vertical  plane  intersects 
the  front,  back,  top,  and  bottom  planes  in  a  rectangle  whose  sides 
develop  into  a  vertical  line  when  the  planes  are  revolved  into  the 
plane  of  the  front  plane.  Thus  the  views  AT,  AB,  and  AG  are  in  a 
vertical  line  passing  through  AF. 


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FIG.  22. 

ioo.  Position  of  a  Point  (Figs.  21  and  22).  Point  ,4  is  a  cer- 
tain distance  from  each  of  the  planes  of  projection.  In  the  front 
view  AF—i  is  the  distance  of  A  above  the  bottom  plane ;  AF—2  is  the 
distance  of  A  below  the  top  horizontal  plane  ;  AF-j  and  AF~4  are  the 
distances  of  A  from  the  side  vertical  planes.  In  the  top  view  the  re- 
lations of  A  to  the  side  vertical  planes  are  seen,  also  the  distance 
AT-2,  of  A  behind  the  front  vertical  plane.  Projecting  lines  from 
AT  to  the  right  and  left  horizontal  axes  give  the  top  projections  A', 
A"  of  A  upon  the  side  vertical  planes.  When  the  left  and  right 
horizontal  axes  are  revolved  about  points  j  and  6  respectively,  points 


ORTPrOGRAPHIC  PROJECTION.  73 

A"1  and  A""  are  obtained.  The  side  views  of  A  are  in  verticals  from 
these  points  and  in  a  horizontal  line  through  AF.  The  distance  AL-j 
and  AR~4  is  thus  the  same  as  AT-2,  and  shows  the  distance  of  A  from 
the  front  vertical  plane.  In  the  same  way  the  view  of  A  upon  the 
bottom  plane  agrees  with  the  other  views. 

101.  Having  the   front   and  top  views  of  any  object  or  point 
given,  the  side  views  can  always  be  obtained  by  projecting  from  the 
top  view  to  the  left  and  right  horizontal  axes,  revolving  these  axes 
until  they  coincide  with  the  front  horizontal  axis,  then  drawing  verticals 
from  the  points  in  the  horizontal  axes  and  intersecting  them  by  hori- 
zontals from  the  points  of  the  front  view.    Having  the  front  and  side 
views  given,  the  reverse  of  the  above  process  will  give  the  top  or 
bottom  view.     When  the  planes  are  seen  from  above,  and  the  points 
of  the  top  view  have  been  projected  to  the  side  planes  (that  is  to  the 
right  and  left  horizontal  axes)  these  projected  points  describe,  when 
the  planes  (axes)  revolve,  arcs  of  circles  whose  centres  are  in  the  ends 
of  the  axes  (5  or  6,  Fig.  22.)     In  the  same  way,  the  points  of  either 
side  view,  when  they  have  been  projected  to  the   horizontal  axis, 
describe,  when  the  axis  revolves,  arcs  of  circles  whose  centre  is  an 
end  of  the  axis. 

102.  In  order  to  understand  the  subject,  students  must  accus- 
tom themselves  to  looking  at  the  different  views  separately.     When 
looking  at  AF  only  the  front  view  should  be  seen.     The  lower  edge 
of  the  front  plane  should  represent  to  their  minds  the  bottom  plane, 
which  seen  edgewise  appears  a  line ;  the  upper  edge  of  the  front 
plane  should  represent  the  top  plane,  of  which  the  edge  only  is  seen, 
and  the  vertical  edges  of  the  front  plane  should  represent  the  vertical 
side  planes,  which,  seen  edgewise,  appear  vertical  lines. 

In  the  same  way,  when  the  top  plane  or  a  side  plane  is  seen,  the 
edges  of  these  planes  should  represent  the  planes  at  right  angles  to 
them.  It  is  possible  to  memorize  rules  and  methods  and  in  this  way 
to  make  drawings  which  are  correct;  but  this' does  not  give  the 
power  to  do  original  work,  or  the  best  work,  or  to  attain  real  under- 
standing of  the  subject.  The  lines  must  represent  planes  and  solids 
and  space,  and  only  by  means  of  -this  mental  construction  of  the 
actual  conditions,  can  drawings  mean  more  than  the  lines  of  a  com- 
plicated plane  geometrical  construction. 


74 


MECHANICAL    DRAWING. 


103.  Views  of  a  Straight  Line.  Fig.  23  represents  the  planes 
of  projection  and  a  triangular  prism  formed  by  bisecting  a  cube  as 
explained  in  Art.  27.  The  edges  of  this  object  represent  the  lines  to 
be  studied. 

The  views  are  obtained  by  means  of  projecting  lines,  which  are 
perpendicular  to  the  planes.  It  follows  that  a  line  which  is  perpen- 


LEfT 


FIG.  33. 

dicular  to  any  plane  will  be  projected  upon  that  plane  as  a  point,  for 
the  projecting  lines  from  its  two  ends  coincide  with  each  other  and 
with  the  line.  Thus  points  2  and  j  are  the  upper  ends  of  vertical 
edges  which  are  perpendicular  to  the  top  plane  and  are  represented 
in  the  top  view  by  points  2  and  j;  points  /  and  4  are  the  back  ends 
of  edges  which  are  perpendicular  to  the  front  plane  and  are  projected 


ORTHOGRAPHIC  PROJECTION. 


75 


upon  it  as  points  i  and  4,  Edges  2-3  and  1-4  are  perpendicular  to 
the  side  planes  and  are  represented  in  the  side  views  by  points  2,  3 
and  7,  4. 

The  projecting  lines  to  any  one  plane  are  parallel  to  each  other; 
the  distance  between  them  must  be  measured  perpendicularly,  that 
is  by  a  line  which  is  parallel  to  the  plane  to  which  the  projecting 
lines  extend.  It  follows  that  the  view  of  a  line  on  a  plane  to  which 
it  is  parallel  must  be  parallel  and  equal  to  the  line.  Edges  2-3  and 
1-4  are  parallel  to  both  top  and  front  planes  and  their  views  upon 
these  planes  are  thus  parallel  and  equal  to  each  other  and  to  the 
lines. 

Edge  1-2  is  oblique  to  the  top  and  front  planes,  therefore  the  pro- 
jecting lines  from  its  ends  to  these  planes  make  its  views  shorter 
than  its  actual  length.  The  edge  1-2  is  parallel  to  the  side  planes,  so 
its  views  upon  these  planes  give  its  real  length. 

A  line  connecting  the  corners  i  and  3  is  a  diagonal  of  the 
sloping  face  of  the  prism ;  it  is  oblique  to  all  the  planes  and  there- 
fore in  all  the  views  the  distance  1-3  is  less  than  the  actual  distance 
between  the  points  i  and  3. 


J. 

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FIG.  24. 


FIG.  25. 


104.  Views  of  a  Plane  Surface.  Figs.  24  and  25  represent 
three  planes  of  projection,  and  a  rectangular  card,  i,  2,  3,  4,  parallel 
to  the  front  vertical  plane  and  perpendicular  to  the  top  and  side 
planes.  Those  who  understand  the  previous  figures  will  understand 
these  drawings  at  a  glance.  The  front  view  gives  the  real  dimen- 


76 


MECHANICAL   DRAWING. 


sions  of  the  card  and  the  positions  of  its  edges  with  reference  to  the 
top  and  side  planes  ;  the  top  view  gives  the  width  of  the  card,  its 
relation  to  the  front  and  side  planes,  and  its  distances  from  these 
planes ;  the  side  view  gives  the  length  of  the  card,  its  relation  to  the 
front  and  top  planes,  and  its  distances  from  these  planes. 

105.  Views  of  a  Solid.  —  Figs.  26  and  27  represent  a  rectan- 
gular pyramid  whose  axis  is  vertical  and  at  given  distances  from  the 
planes  of  projection.  Two  edges  of  the  base  of  the*pyramid  are 
parallel  and  two  are  perpendicular  to  the  front  plane. 

Fig.  26  is  a  perspective  view  which  represents  the  pyramid  and 
the  planes  of  projection,  with  the  views  of  the  pyramid  upon  them. 


FIG.  26. 


FIG.  27. 


Fig.  27  represents  the  planes  and  their  respective  views  when  the 
top  and  side  planes  have  been  revolved  into  the  plane  of  the  front 
plane. 

The  base  of  the  pyramid  is  at  right  angles  to  the  front  plane, 
upon  which  it'  is  projected  as  a  horizontal  line  IF-2F  ;  it  is  parallel  to 
the  top  plane,  upon  which  its  real  shape  is  given. 

The  axis,  being  perpendicular  to  the  top  plane,  is  projected  upon 
it  as  a  point  jr,  at  the  centre  of  the  base,  and  to  this  point  the  top 
views  of  the  lateral  edges  extend  and  form  the  diagonals  of  the 
figure.  The  right  and  left  triangular  faces  of  the  pyramid  are  per- 
pendicular to  the  front  plane,  and  are  projected  upon  it  as  lines 
2F-$F  and  IF-_5F.  The  two  wide  triangular  faces  of  the  pyramid 
are  oblique  to  the  front  plane,  and  the  two  narrow  triangular  faces 


OR THOGRAPffIC  PROJECTION. 


77 


are  oblique  to  the  side  plane  ;  thus  th£  real  shapes  of  these  faces  are 
not  given  in  either  view. 

106.  Simple  geometric  forms  are  used  in  the  first  study  of  projec- 
tion.    Generally  the  views  of  these  forms  upon  the  front,  top,  and  one 
side  plane  are  all  that  are  required  to  describe  the  object.     It  is  not 
customary  to  limit  these  planes  of  projection  except  by  the  three 
lines,  or  axes,  in  which  they  intersect.     These  axes  are  the  front 
horizontal  axis,  a  left  or  right  horizontal 

axis,  and  a  vertical  axis  at  the  left  or  right, 
and  are  represented  in  the  illustration. 

The  F.  hor.  axis  is  the  only  one  re- 
quired to  locate  objects  with  reference  to 
the  front  and  top  planes.  Figs.  28,  29, 
and  30  show  how  the  views  of  objects 
may  be  obtained,  using  only  this  axis. 

107.  Views  of  a  Rectangular  Card.  —  Fig.  28  gives  the  front 
and  top  views  of  a  rectangular  card,  2f'X4",  which  is  parallel  to,  and 


ffonf 


Tiar.f/is. 


J  . 


FIG.  28. 


FIG.  30. 


2"  behind  the  front  plane  ;  its  short  edges  are  parallel  to  the  top 
plane,  the  upper  one  being  i"  below  the  plane. 

The  front  view  is  a  rectangle  2lfX4",  its  long  sides  vertical, 
and  its  upper  short  side  i"  below  the  axis  of  projection.  The  top 
view  is  a  horizontal  line  2"  above  the  axis  of  projection. 


78  MECHANICAL   DRAWING. 

1 08.  Fig.  29  represents  tHte  same  card  as    Fig.  28,  when  it  is 
parallel  to  the  front  plane,  and  the  same  distance  behind  it  as  in 
Fig.  28,  but  with  its  long  edges  at  60°  to  the  top  plane.     The  front 
view  gives  the  real  shape  of  the  card,  and  the  real  angles  which 
its  edges  make  with  the  top  plane.     The  top  view  is  a  horizontal 
line. 

109.  Fig.  30  represents  the  same  card  when  perpendicular  to 
the  top  plane,  and  the  same   distance  from  it  as  in   Fig.   29  ;  its 
edges  are  at  60°  and  30°  to   the  top  plane  as  in  Fig.  29,  but  its 
surface  is  at  30°  to  the  front  plane,  instead  of  parallel  to  the  plane 
as  in  Fig.  29. 

When  the  card  is  in  the  position  shown  by  Fig.  29,  its  top  view 
is  a  horizontal  line  whose  length  is  obtained  by  projecting  from  2 
and  j  of  the  front  view.  If  the  card  is  revolved  about  a  vertical  axis 
passing  through  point  i,  the  angles  of  its  edges  and  its  surface  with 
the  top  plane  not  changing,  the  top  view  will  show  the  angle  of  the 
card  with  the  front  plane,  but  its  length  will  not  change ;  therefore, 
for  all  positions  of  the  card  during  a  complete  revolution,  the  only 
change  in  the  top  view  is  in  the  angle  which  it  makes  with  the  front 
horizontal  axis. 

Suppose  the  card  to  revolve  about  point  i  as  described,  points  2, 
j,  and  4  describe  horizontal  circles  whose  centres  are  in  a  vertical 
line  passing  through  i.  These  circles  appear  circles  in  the  top 
view,  and  as  horizontal  lines  in  the  front  view.  It  will  be  seen  that 
points  .2,  j,  and  4  of  the  front  view,  which  represents  the  card  when 
its  surface  is  at  an  angle  to  the  front  plane,  must  be  in  horizontal 
lines  drawn  through  the  corresponding  points  of  Fig.  29.  Hence,  to 
obtain  the  front  view  of  the  card  when  at  any  given  angle,  it  is 
necessary  simply  to  place  the  top  view  of  Fig.  29  at  the  required 
angle  and  draw  vertical  projecting  lines  from  all  its  points,  and 
intersect  them  by  horizontals  from  the  corners  of  the  front  view  of 
Fig.  29. 

The  views  of  Fig.  30  cannot  be  obtained  without  making  use  of 
those  of  Fig.  29,  or  of  substitute  drawings.  The  views  of  a  solid 
oblique  to  one  or  both  the  principal  planes  of  projection  may  be 
determined  in  the  same  way  as  those  of  the  card. 

no.    Views    of   a   Pyramid Fig.  31    represents    a  pyramid 


ORTHOGRAPHIC  PROJECTION. 


79 


whose  base  is  a  rectangle  2^X3";  its  axis  is  5"  long,  is  vertical, 
and  2"  behind  the  front  plane  ;  the  vertex  of  the  pyramid  is  i"  from 
the  top  plane,  and  the  long  edges  of  the  base  are  parallel  to  the  front 
plane. 

in.  Fig.  32  represents  the  same  pyramid  after  it  has  been 
revolved  to  the  right  through  an  angle  of  45°,  about  the  right  edge 
of  the  base  ;  all  the  edges  of  the  object  are  situated  with  reference 
to  the  front  plane  as  they  are  in  Fig.  31. 

Suppose  the  object  to  be  moved  to  the  right  of  its  position  in 
Fig.  31,  its  distance  from  the  front  plane  not  changing,  and  then 


FIG.  31. 


FIG.  32. 


FIG.  33. 


FIG.  34. 


revolved  to  the  right  about  the  edge  2-4.  As  the  object  revolves, 
all  points  in  it  move  in  arcs  of  circles  whose  centres  are  in  2-4  and 
which  are  parallel  to  the  front  plane  and  appear  horizontal  lines  in 
the  top  view.  The  points  in  the  top  view  of  Fig.  32  must  then  be 
in  horizontal  lines  drawn  through  the  corresponding  points  of  the  top 
view  of  Fig.  31. 

The  object  may  revolve  through  a  complete  circle,  and  cause  no 
change  in  the  size  or  shape  of  the  front  view;  thus  the  front  view  of 
Fig.  32  will  be  the  front  view  of  Fig.  31,  with  the  axis  at  an  angle  of 
45°  instead  of  vertical.  The  top  view  will,  however,  present  an 
infinite  number  of  different  appearances,  as  the  object  revolves. 


8O  MECHANICAL   DRAWING. 

To  obtain  the  views  of  the  object  when  its  axis  is  parallel  to  the 
front  plane  as  in  Fig.  31,  and  at  any  angle  to  the  top  plane,  it  is 
simply  necessary  to  place  the  front  view  of  Fig.  31  with  its  axis  at 
the  required  angle  ;  to  draw  vertical  projecting  lines  from  all  its 
corners  and  intersect  these  verticals  by  horizontals  from  the  corre- 
sponding points  of  the  top  view  of  Fig.  31.  The  front  view  in  Fig. 
32  might  be  drawn  without  reference  to  Fig.  31,  but  it  will  often  be 
necessary  to  draw  the  views  of  an  object  situated  as  in  Fig.  31,  in 
order  that  the  shape  of  the  front  view  for  Fig.  32  may  be  known. 

112.  Fig.  33  represents  the  object  when  its  axis  is  at  an  angle 
with  both  planes. 

Suppose  the  pyramid  placed  as  in  Fig.  32,  with  its  axis  at  45°  to 
the  top  plane,  to  be  moved  horizontally  to  the  right  and  then  revolved 
about  a  vertical  axis  passing  through  its  vertex,  through  a  complete 
circle,  the  angle  of  the  axis  with  the  top  plane  not  changing,  and 
the  distances  of  all  points  of  the  object  from  the  top  plane  remaining 
as  in  Fig.  32.  The  top  view  of  the  pyramid  in  any  and  all  of  its 
positions  must  be  the  same  in  form  and  size  as  the  top  view  in  Fig. 
32.  Hence,  to  obtain  the  projections  of  the  object  when  its  axis  is 
at  45°  to  the  top  plane,  as  in  Fig.  32,  and  the  vertical  plane  con- 
taining the  axis  is  at  any  given  angle,  say  45°,  to  the  front  plane,  we 
have  simply  to  place  the  top  view  of  Fig.  32  so  that  the  axis  of  the 
pyramid  makes  an  angle  of  45°  to  the  front  axis  of  projection,  and 
draw  verticals  from  the  points  of  this  view,  and  intersect  them  by 
horizontals  from  the  corresponding  points  of  the  front  view  of  Fig. 
32.  Thus  the  front  view  of  any  point,  as  i,  will  be  in  a  vertical 
from  z  in  the  top  view  of  Fig.  33,  and  in  a  horizontal  from  i  of  the 
front  view  of  Fig.  32. 

In  this  way,  by  considering  one  point  at  a  time,  the  views  of  the 
most  complicated  objects  can  be  obtained. 

Similarly,  the  front  view  of  Fig.  33  may  be  revolved  through  any 
desired  angle  and  a  top  view  to  correspond  be  found  by  projecting 
from  the  top  view  of  Fig.  33  ;  this  process  of  revolving  first  one  view 
and  then  the  other  may  be  repeated  as  many  times  as  is  desired. 

Generally  no  more  than  the  revolution  of  one  top  view  and 
one  front  view  will  be  necessary  to  give  views  of  any  object  at  angles 
to  both  planes  of  projection. 


ORTHOGRAPHIC  PROJECTION.  8 1 

» 

Fig.  34  shows  the  top  view  of  Fig.  31  revolved  through  an  angle 
of  45°.  The  front  view  in  Fig.  34  may  be  revolved  as  desired  and 
thus  any  appearance  of  the  object  be  obtained. 

113.  All  the  vertical   and  horizontal  projecting  lines  may  be 
drawn  before  any  of  the  points  of  the  required  view  are  determined  ; 
but  it  is  much  simpler  for  elementary  pupils  to  draw  the  projecting 
lines  for  the  different  points,  one  at  a  time,  and  mark  the  points  of 
the  view  as  they  are  determined. 

The  points  of  solids  studied  should  be  numbered,  and  the  num- 
bers carefully  placed  in  all  the  views  so  that  they  indicate  jcor- 
responding  points.  When  this  has  been  done,  to  obtain  the  points 
for  any  required  view,  as  the  top  view  of  Fig.  32,  it  is  simply  neces- 
sary to  draw  a  vertical  from  any  point,  as  j",  of  the  front  view  and  in- 
tersect it  by  a  horizontal  from  5  of  the  top  view  of  Fig.  3 1 . 

114.  True  Length  and  Position  of  a  Straight  Line.    The  true 
length  of  a  line  is  given  by  any  view  only  when  the  line  is  parallel  to 
the  plane  upon  which  the  view  is  made  ;  when  thus  situated  the  view 
gives  not  only  the  true  length  of  the  line,  but  its  true  positions  with 
reference  to  the  planes  at  right  angles  to  the  one  upon  which  the 
view  is  made.     Frequently  a  line  is  oblique  to  all  the  planes,  and  its 
real  length  is  not  shown  by  any  view  ;  hence  it  is  important  that  a 
simple  method  of  determining  the  real  length  of  a  line  be  found. 

This  problem  is  solved  in  the  figures  of  this  book  as  follows : 
Drawing  A  represents  a  line  ab  which  is  situated  so  that  its 
front  view  makes  an  angle  of  30°  with  the  top  plane  (F.  H.  axis), 
and  its  top  view  makes  an  angle  of  45°  with  the  front  plane  (F.  H. 
axis).  To  find  the  real  length  of  a  b  and  the  angle  it  makes  with 
the  top  plane,  the  line  must  be  revolved  until  it  is  parallel  to  the 
front  plane,  when  it  will  be  represented  in  the  top  view  \>y  ATb'. 
As  a  b  revolves,  b  moves  in  an  arc  which,  in  the  top  view,  is  repre- 
sented by  the  arc  b'r  b\  and  in  the  front  view  by  the  horizontal  line 
bFb".  Line  aF b"  is  the  real  length  of  a  b,  and  gives  the  real  angle 
of  a  b  with  the  top  plane. 

If  the  angle  of  ab  with  the  front  plane  is  desired,  ab  must  be 
revolved  until  it  is  parallel  to  the  top  plane,  as  illustrated  by  drawing 
B.  As  a  b  revolves,  b  moves  in  an  arc  which  is  represented  in  the 
front  view  by  the  arc  bF b' ;  and  in  the  top  view  by  the  horizontal 


82 


MECHANICAL   DRAWING. 


ABC 

line  bTb".  Line  aT b"  is  the  real  length  of  ab,  and  gives  its  real 
angle  with  the  front  plane. 

In  drawing  A,  the  real  length  of  a  b  is  equal  to  the  hypotenuse  of  a 
right-angled  triangle,  of  which  the  base  is  equal  to  the  length  of  the  top 
vieu>  of  a  b,  and  the  altitude  is  equal  to  the  difference  in  the  distances  of 
points  a  and  b  from  the  top  plane. 

In  drawing  B,  the  real  length  of  a  b  is  equal  to  the  hypotenuse  of  a 
right-angled  triangle  whose  base  is  equal  to  the  length  of  the  front  view 
o/a.b,  and  whose  altitude  is  equal  to  the  difference  in  the  distances  of 
points  a  and 'b  from  the  front  plane. 

The  real  lengths  of  a  number  of  lines  of  different  lengths  will  be 
obtained  most  quickly  by  constructing,  at  any  convenient  place  out- 
side the  drawing,  right-angled  triangles  according  to  the  above  para- 
graphs ;  these  triangles  may  have  a  common  right  angle. 

115.  When,  instead  of  both  views  being  given  and  the  true 
length  and  angles  being  required,  the  angle  of  one  view,  and  the  real 
length  of  the  line,  and  its  real  angle  with  the  other  plane  are  given, 
the  views  of  the  line  will  be  found  as  follows : 

Suppose  that  a  b  is  the  length  aF  b"  (drawing  A)  and  makes  the 
angle  m  with  the  top  plane,  while  its  top  view  makes  the  angle  45° 
with  the  front  plane.  Draw  first  aF b"  of  the  proper  length,  and  at 
the  angle  m  with  the  top  plane  ;  then  project  and  obtain  a'r b\  which 
is  the  length  of  the  top  view.  From  aT  draw  the  top  view  at  the 
given  angle,  45°,  and  make  its  length  equal  to  a7* b'  by  drawing  the 
arc  b'  bT.  Project  from  br  to  the  level  of  b"  and  obtain  bF,  join  bF 
and  aF  and  the  front  view  is  given. 


ORTHOGRAPHIC  PROJECTION.  83 

The  views  will  be  obtained  in  the  same  way  if  a  b  is  the  length 
aT b"  (drawing  B},  and  at  the  angle  n  with  the  front  plane,  and  hav- 
ing its  front  view  at  the  angle  30°  with  the  top  plane. 

1 1 6.  Drawing  C  illustrates  the  way  in  which  the  views  of  a  line 
are  found,  when,  instead  of  the  angles  of  the  views,  the  real  angles 
with  both  planes  and  the  real  length  of  the  line  are  given. 

Line  cd  is  the  distance  CF  d'  in  length  ;  it  makes  an  angle  of 
30°  with  the  top  plane,  and  one  of  45°  with  the  front  plane.  To 
obtain  the  views,  whose  angles  are  not  known,  draw  first  CF  d1  equal 
in  length  to  cd  and  at  the  angle  30°  with  the  top  plane ;  then  draw 
CT d"  equal  in  length  to  cd,  and  at  the  angle  45°  with  the  front 
plane.  From  CF d'  project  to  obtain  crd"',  which  is  the  real  length 
of  the  top  view  of  cdvfhen  it  is  at  the  angle  30°  with  the  top  plane. 
From  CT d"  project  to  obtain  cFd"",  which  is  the  length  of  the  front 
view  of  <r</when  it  is  at  the  angle  45°  with  the  front  plane.  Point 
d  must  be  as  far  from  the  top  plane  as  is  d',  and  as  far  from  the 
front  plane  as  is  d" ;  hence,  dF  must  be  in  an  arc  whose  radius  is 
cFd"",  and  in  a  horizontal  line  through  d' ;  and  point  dT  must  be  in 
an  arc  whose  radius  is  cTd"',  and  in  a  horizontal  line  through  d". 

From  the  preceding  pages,  the  following  principles  may  be 
noted.  They  are  also  illustrated  by  the  drawings  of  the  previous 
chapters. 

VIEWS  OF  A  POINT. 

117.  Any  view  of  a  point  is  a  point. 

118.  The  top,  front,  and  bottom  views  of  a  point  are  in  the  same 
vertical  line. 

119.  The  front,  side,  and  back  views  of  a  point  are  in  the  same 
horizontal  line. 

VIEWS  OF  A  STRAIGHT  LINE. 

120.  Any  view  of  a  straight  line  is  a  line  or  a  point. 

121.  Any  view  of  a  line  on  a  plane  to  which  it  is  parallel  is  par- 
allel and  equal  to  the  line. 

122.  Any  view  of  a  line  on  a  plane  to  which  it  is  perpendicular 
is  a  point. 

123.  Any  view  of  a  line  on  a  plane  to  which   it  is  oblique  is 
shorter  than  the  line  itself, 


84  MECHANICAL   DRAWING. 

124.  Parallel  and  equal  lines  are  represented  on  any  plane  by 
parallel  and  equal  lines. 

125.  A  straight  line  parallel  to  the  front  plane  and  perpendicular  to 
the  top  plane. 

The  front  view  is  a  vertical  line  whose  length  equals  that  of  the 
line ;  the  top  view  is  a  point ;  either  side  view  is  a  vertical  line  whose 
length  equals  that  of  the  line. 

126.-  A  straight  line  parallel  to  the  front  plane  and  perpendicular  to 
the  side  planes. 

The  front  view  is  a  horizontal  line  whose  length  equals  that  of  the 
line  ;  the  top  view  is  a  horizontal  line  whose  length  equals  that  of  the 
line  ;  either  side  view  is  a  point. 

127.  A  straight  line  parallel  to  the  front  plane  and  oblique  to  the  top 
and  side  planes. 

The  front  view  is  an  oblique  line  which  gives  the  real  length  and 
the  angles  of  the  line  with  the  top  and  side  planes ;  the  top  view  is 
a  horizontal  line  shorter  than  the  line  itself ;  either  side  view  is  a  ver- 
tical line  shorter  than  the  line  itself. 

128.  A  straight  line  perpendicular  to  the  front  plane. 

The  front  view  is  a  point ;  the  top  view  is  a  vertical  line  whose 
length  equals  that  of  the  line  itself  •  either  side  view  is  a  horizontal 
line  whose  length  equals  that  of  the  line  itself. 

129.  A  straight  line  oblique  to  the  front  plane  and  parallel  to  the  top 
plane. 

The  front  view  is  a  horizontal  line  shorter  than  the  line  itself; 
the  top  view  is  an  oblique  line  which  gives  the  real  length  and  the 
angles  of  the  line  with  the  front  and  side  planes ;  either  side  view  is 
a  horizontal  line  shorter  than  the  line  itself. 

130.  A  straight  line  oblique  to  the  front  plane  and  parallel  to  the 
side  planes. 

The  front  view  is  a  vertical  line  shorter  than  the  line  itself ;  the 
top  view  is  a  vertical  line  shorter  than  the  line ;  either  side  view  is 
an  oblique  line  which  gives  the  real  length  of  the  line  and  the  angles 
of  the  line  with  the  front  and  top  planes. 

131.  A  straight  line  oblique  to  the  front,  top,  and  side  planes. 

The  front,  top,  or  side  view  is  an  oblique  line  shorter  than  the 
line  itself. 


ORTHOGRAPHIC  PROJECTION.  85 


VIEWS  OF  PLANE  SURFACES  OR  FIGURES. 

132.  Any  view  of  a  plane  surface  or  figure  is  a  figure  or  a  line. 

133.  A  plane  surface  or  figure  parallel  to  the  front  plane. 

The  front  view  gives  the  real  size  and  shape  of  the  figure,  and 
the  real  angles  that  its  edges  make  with  the  top  and  side  planes  ; 
the  top  view  is  a  horizontal  line  ;  and  either  side  view  is  a  vertical  line. 

134.  A  plane  surface  or  figure  perpendicular  to  the  front  plane  and 
parallel  to  the  top  Qlane. 

The  front  view  is  a  horizontal  line ;  the  top  view  gives  the  real 
size  and  shape  of  the  figure  and  the  real  angles  that  its  edges  make 
with  the  front  and  side  planes ;  and  either  side  view  is  a  horizontal 
line. 

I35«  A  plane  surface  or  figure  perpendicular  to  the  front  plane  and 
parallel  to  the  side  planes. 

The  front  view  is  a  vertical  line  ;  the  top  view  is  a  vertical  line ; 
either  side  view  gives  the  real  size  and  shape  of  the  figure  and  the 
real  angles  that  its  edges  make  with  the  front  and  top  planes. 

136.  A  plane  surface  or  figure  perpendicular  to  the  front  plane  and 
oblique  to  the  top  and  side  planes. 

The  front  view  is  an  oblique  line  which  shows  the  real  angles  of 
the  plane  with  the  top  and  side  planes  ;  the  top  view  is  a  figure 
which  is  foreshortened  in  one  direction  ;  either  side  view  is  a  figure 
which  is  foreshortened  in  one  direction. 

137.  A  plane  surface  or  figure  oblique  to  the  front  and  side  planes 
and  perpendicular  to  the  top  plane. 

The  front  view  is  a  figure  which  is  foreshortened  in  one  direc- 
tion ;  the  top  view  is  an  oblique  line  which  shows  the  real  angles  of 
the  plane  with  the  front  and  side  planes  ;  either  side  view  is  a  figure 
which  is  foreshortened  in  one  direction. 

138.  A  plane  surface  or  figure  oblique  to  the  front  and  top  planes 
and  perpendicular  to  the  side  planes. 

The  front  view  is  a  figure  which  is  foreshortened  in  one  direc- 
tion ;  the  top  view  is  a  figure  which  is  foreshortened  in  one  direction  ; 
either  side  view  is  an  oblique  line  which  shows  the  real  angles  of  the 
plane  with  the  front  and  top  planes. 


86  MECHANICAL    DRAWING. 

139.  A  plane  surface  or  figure  all  of  whose  edges  are  oblique  to  the 
front,  top,  and  side  planes. 

The  front,  top,  or  either  side  view  is  a  figure  all  of  whose  lines 
are  shorter  than  the  lines  of  the  figure  they  represent. 

140.  Parallel  and  equal  plane  surfaces  or  figures,  whose  corre- 
sponding edges  are  parallel,  are  represented  on  any  plane  by  similar 
figures  whose  corresponding  lines  are  equal  and  parallel. 

LOCATIONS  OF  THE  VIEWS. 

141.  The  top  and  bottom  views  are  always  respectively  above 
and  below  the  front  view. 

142.  The  side  views  are  always  on  the  same  level  as  the  front 
view. 

143.  The  view  of  the  right  side  is  always  at  the  right  of  the 
front  view,  and  the  view  of  the  left  side  is  always  at  the  left  of  the 
front  view. 

144.  The  line  of  any  view  which  is  nearest  the  front  view  repre- 
sents the  front  face  or  front  line  of  the  object. 

PROJECTION    PROBLEMS. 

These  problems  are  arranged  for  those  who  wish  to  work  by  pro- 
jection methods.  They  are  not  intended  for  public  school  pupils. 

All  polygons  and  solids  referred  to  are  regular,  unless  otherwise 
stated.  When  two  views  are  asked  for,  the  top  and  front  views  are 
desired. 

1.  Two  views  of  a  line  2"  long.     It  is  parallel  to  the  top  plane 
and  i"  below  it,  and  at  45°  to  the  front  plane,  with  its  nearest  end 
J^  "  behind  the  front  plane. 

2.  Two  views  of  a  line  3"  long.     It  is  parallel  to  the  front  plane, 
and  Jis"  behind  it,  and  at  60°  to  the  top  plane,  from  which  its  upper 
end  is  y2  "  distant. 

3.  Two  views  of  a  square  card  whose  edges  are  2"  long.     It  is 
parallel   to  the  front  plane  and  has  two  edges  vertical.     It  is  ^" 
behind  the  front  plane,  and  its  upper  edge  is  %"  below  the  top 
plane. 


PROJECTION  PROBLEMS.  87 

• 

4.  Two  views  of  a  circular  card  2"  in  diameter.     It  is  parallel  to 
the  front  plane,  with  its  centre  i"  behind  the  front  plane  and  i^" 
below  the  top  plane. 

5.  Two  views  of  the  same  card,  when  it  is  parallel   to  the  top 
plane,  its  centre  being  il/2"  behind  the  front  plane  and  y2"  below  the 
top  plane. 

6.  Two  views  of  an  equilateral  triangular  card  whose  edges  are 
2"  long.    It  is  parallel  to  the  front  plane  ;  its  lowest  edge  is  horizon- 
tal and  2^2"  below  the  top  plane  and  ^"  behind  the  front  plane. 

7.  (#)  Two  views  of  an  hexagonal  card  whose  edges  are  i"  long. 
It  is  parallel  to  the  front  plane  and  ^"  behind  it,  with  two  edges 
horizontal  and  the  upper  one  ^  "  below  the  top  plane. 

(£)  Two  projections  of  the  same  card  in  the  same  position  with 
reference  to  the  top  plane,  but  at  45°  to  the  front  plane,  the  nearest 
point  being  %"  behind  the  front  plane. 

8.  Two  views  of  a  prism  4"  long,  whose  bases  are  squares  of  2". 
The  long  edges  are  vertical.     The  upper  base  is  */2 "  below  the  top 
plane,  and  the  nearest  lateral  face  is  parallel  to  and  i"  behind  the 
front  plane. 

9.  Two  views  of  a  cone  whose  axis  is  4"  long  and  whose  base  is 
2"  in  diameter.     Its  axis  is  vertical  and  1^2"  behind  the  front  plane, 
the  vertex  is  ^"  below  the  top  plane. 

10.  Two  views  of  a  pyramid  whose  axis  is  4"  long,  and  whose 
base  is  a  square  of  2".     The  axis  is  vertical  and  3"  behind  the  front 
plane;  its  upper  end  is  i"  below  the  top  plane.     The  edges  of  the 
base  are  at  45°  to  the  front  plane. 

11.  Two  views  of  an  hexagonal  prism  4"  long.     The  prism  is 
vertical,  its  nearest  lateral  face  being  parallel  to  and  ^  "  behind  the 
front  plane.     The  edges  of  its  bases  are  i"  long,  and  its  upper  base 
is  1 1^"  below  the  top  plane. 

12.  Two  views  of  an  hexagonal  pyramid  of  the  same  dimensions 
as  the  prism  of  Problem  n,  the  centre  of  the  base  being  5^"  below 
the  top  plane  and  2^"  behind  the  front  plane.      A  long  diagonal 
of  the  base  is  parallel  to  the  front  plane. 

13.  Two  views  of  a  triangular  prism  4"  long,  whose  bases  are 
equilateral  triangles  of  2"  sides,  when  the  lateral  edges  are  horizontal 
and  parallel  to  the  front  plane.     One  lateral  face  is  horizontal   and 


88  MECHANICAL    DRAWING. 

3"  below  the  top  plane.     The  prism  extends  above  this  face,  whose 
nearest  edge  is  ^  "  behind  the  front  plane. 

14.  (a)  Two  views  of  a  prism  4"  long,  whose  bases  are  squares 
of  2"  side,  when  its  lateral  edges  are  parallel  to  the  front  plane  and 
at  60°  to  the  top  plane.     Two  lateral  faces  are  parallel  to  the  front 
plane  ;  the  nearer  one  is  3"  behind  the  front  plane,  and  its  upper 
corner  is  %"  below  the  top  plane. 

(£)  The  same  prism,  the  same  distance  below  and  at  the  same 
angle  with  the  top  plane  ;  but  its  vertical  faces  at  30°  to  the  front 
plane.  The  nearest  corner  of  the  object  is  y2"  behind  the  front 
plane. 

15.  (a)  Two  views  of  a  pyramid  4"  long,  whose  base  is  a  square 
of  2".     The  centre  of  the  base  is  2^"  behind  the  front  plane  and 
6j4"  below  the  top  plane,  and  the  axis  of  the  pyramid  extends  to  the 
right  parallel  to  the  front  plane  and  upward  at  45°  to  the  top  plane. 
A  vertical  plane  through  the  axis  contains  two  of  the  lateral  edges  of 
the  object. 

(fr)  Project  the  pyramid  upon  the  right  side  plane,  which  is  i "  to 
the  right  of  the  vertex  of  the  pyramid. 

1 6.  Two  views  of  a  cylinder  4"  long  and  2"  in  diameter,  whose 
axis  is  parallel  to  the  front  plane,  at  45°  to  the  top  plane  and  i^" 
behind  the  front  plane.    The  upper  end  of  the  axis  is  2^"  below  the 
top  plane. 

17.  (a)  Two  views  of  a  cone  whose  axis  is  4"  long,  and  whose 
base  is  2"  in  diameter.    The  lowest  element  of  the  cone  is  horizontal 
and  parallel  to  the  front  plane  ;  it  is  2^2  "  behind  the  front  plane  and 
3  yz  "  below  the  top  plane. 

($)  Two  views  of  the  same  cone  at  the  same  level,  when  the  end 
of  the  horizontal  element  at  the  base  of  the  cone  is  i"  behind  the 
front  plane,  the  element  extending  from  this  point  to  the  right  from 
the  front  plane  at  45°  with  it. 

1 8.  The  front  view  of  a  line  is  2"  long  and  is  at  60°  to  the  front 
axis  of  projection.    The  top  view  is  at  45°  to  this  axis.    Give  the  real 
length  of  the  line  and  measure  the  actual  angles  made  by  the  line 
with  the  two  planes. 


CHAPTER   IX. 

SECTIONS. 

145.  SUPPOSE  a  plane  to  pass  through  an  object  in  any  direc- 
tion, the  part  of  the  object  in  front  of  the  plane  to  be  removed,  and 
the  projection  of  the  part  of  the  object  behind  the  cutting  plane  to 
be  made  on  a  parallel  plane.     Such  a  projection  is  generally  called 
a  section,  though  section   sometimes   means  a  drawing  which  gives 
simply  the  actual  shape  of  the  section  given  by  the  cutting  plane. 

Sections  are  taken  to  show  the  interior  of  hollow  objects,  or  the 
shape  of  solid  parts  which  are  not  clearly  represented  by  views  of 
the  outside  of  the  object.  Sections  are  most  necessary  and  most 
used  in  practical  working  drawings ;  but  the  principles  involved  can 
best  be  studied  by  the  use  of  simple  geometric  solids. 

146.  Sections  may  be  drawn  in  place  of  the  principal  views,  or 
they  may  be  taken  in  any  direction  whatever.      The  position  of  a 
section  should  be  indicated  by  a  dot-and-dash  line  in  the  view  in 
which  the  plane  of  the  section   is  seen  edgewise  ;  but  if  the  view 
represents  the  object  when  the  part  in  front  of  the  cutting  plane  is 
removed,  the  line  of  the  section  should  be  represented  by  a  full  line 
upon  the  object,  and  by  a  dot-and-dash  line  outside. 

In  practical  working  drawings,  the  objects  are  often  shown  entire 
in  one  view  and  in  section  in  another.  In  the  study  of  projection, 
or  working  drawings,  by  the  use  of  the  geometric  solids,  all  views 
should  agree  in  representing  the  object  with  the  part  in  front  of  the 
cutting  plane  removed.  In  this  case  the  part  removed  may  be 
shown,  if  desired,  by  dotted  or  by  dot-and-dash  lines. 

147.  If  it  is  desired  to  represent  only  the  line  of  intersection 
given  by  passing  any  plane  through  any  solid,  all  views  should  agree 
in  representing  the  entire  solid  with  the  line  of  intersection  upon  its 
surface. 

148.  When  the  part  in  front  of  the  cutting  plane  is  removed,  it 
is  customary  to  show  the  surface  cut  by  the  plane,  by  hatching  it 
with  parallel  equidistant  lines  as  explaiped  in  Art.  84. 


9o 


MECHANICAL    DRAWING. 


149.  Sections  of  the  Sphere.  —  The  simplest 
sections  are  those  of  the  sphere,  for  every  section 
made   by  a   plane    is   a   circle,   whose   diameter 
ranges  from  that  of  the  great  circle,  given  by  a 
plane  passing  through  the  centre  of  the  sphere, 
to  that  of  a  circle  as  small  as  can  be  imagined, 
given  by  a  plane  barely  cutting  the  sphere. 

The  centre  j  of  the  circle  C,  given  by  a  plane 
which  cuts  the  sphere  and  does  not  pass  through 
its  centre,  is  in  a  line  passing  through  the  centre 
of  the  sphere  perpendicular  to  the  cutting  plane. 

150.  Sections  of  the  Cube.  —  Sections  of  a 
cube  are  polygons,  for  the  surfaces  of  the  cube  are  plane,  and  when 
intersected  by  a  plane  must  produce  straight  lines. 

If  a  plane  cuts  three  faces  of  the  cube,  the  section  is  a  triangle  ; 
if  it  cuts  four  faces  the  section  has  four  sides.  The  section  may 
be  a  polygon  of  three,  four,  five,  or  six  sides,  according  to  the  num- 
ber of  faces  which  are  cut  by  the  plane. 

151.  To  obtain  the1  views  of  the  section  given  by  any  cutting 
plane,  the  plane  must  first  be  represented  in  the  view  in  which  it 
appears  a  line,  for  only  in  this  view  are  the  points  in  which  it  inter- 
sects the  edges,  or  elements,  of  the  object  seen.     The  points  of 
intersection,    having 

been  determined  in 
this  view,  are  readily 
projected  to  the  other 
views. 

152.  Suppose   a 
cube,  placed  so  that 
two  vertical  faces  are 
at  45°   to   the  front 
plane,    to    be    inter- 
sected by  a  cutting 
plane   at  30°  to  the 
top  plane  and  perpen- 
dicular to  the  front 
plane. 


SECTIONS. 


The  top  view  of. the  cube  is  a  square  whose  sides  are  at  45°; 
the  front  and  side  views  are  equal  rectangles..  Line  EF  in  the  front 
view  represents  the  cutting  plane.  We  will  suppose  the  part  of  the 
cube  above  this  plane  to  be  removed,  and  will  represent  it  by  dotted 
lines.  In  the  top  view,  the  line  2—3  on  the  top  face  of  the  cube  is 
seen  of  its  true  length  and  is  the  only  edge  of  the  section  which  is 
not  represented  by  the  square.  The  side  view  of  2-3  will  be 
obtained  by  setting  off  one  half  the  distance  2-3  of  the  top  view  each 
side  of  the  centre  of  the  top  line  of  the  side  view,  and  the  other 
points  in  the  section  will  be  obtained  by  projecting  from  the  points, 
in  the  front  view,  where  the  plane  cuts  the  vertical  edges. 

The  cutting  plane  is  oblique  to  the  top  plane  and  also  to  the  side 
plane.  The  section  appears  of  different  shapes  in  these  two  views, 
and  neither  gives  the  real  dimensions  of  the  figure.  Line  i-o  of  the 
top  view  is  the  centre  line  of  the  section  and  is  parallel  to  the  front 
plane ;  therefore  the  front  view  gives  the  real  length  of  the  section 
and  the  distance  of  line  2—3  from  a  line  connecting  4  and  5.  .The 
top  view  gives  the  real  lengths  of  lines  2-3  and  4-5,  and  by  com- 
bining the  lengths  of  the  front  view  with  the  widths  of  the  top  view, 
the  real  shape  of  the  section  will  be  obtained. 

A  simple  way  to  obtain  the  true  shape  is  to  draw  L M  parallel  to 
EF,  and  from  points  i,  2,  and  4  in  EF,  draw  perpendiculars  to  EF; 
these  perpendiculars  in- 
tersecting LM  give  the 
lengths ;  the  widths  are 
obtained  by  setting  off  each 
side  of  LM  the  distances 
of  the  points  in  the  top 
view  from  i-o.  Join  1-4, 
4-2, 2-3,3-5,  and  5-7,  and 
the  real  shape  of  the  sec- 
tion is  obtained. 

153.    The   true   shape 
of     the    section     can     be 
obtained  as  just  explained  5 
only  when  line  i-o  of  the 
top  view  is  parallel  to  the 


92  MECHANICAL   DRAWING. 

front  plane.  When  the  sides  of  the  cube  are  not  at  45°  to  the  front 
plane  the  section  will  not  be  symmetrical,  the  front  view  will  not  give 
the  real  length  of  the  section,  and  line  i— o  will  pass  through  only  one 
of  the  corners  of  the  top  view.  The  real  shape  of  the  section  in  this 
case  can  be  obtained  by  measuring,  in  the  top  view,  the  distances  of 
points  4,  2,  3,  and  3  from  7-0,  by  means  of  perpendiculars  to  7-0, 
and  setting  these  distances  off  on  the  proper  lines,  measuring  from 
LM,  which  as  before  is  drawn  parallel  to  1-2  of  the  front  view  and 
represents  line  7-0  of  the  top  view. 

154.  The  opposite  sides  of  the  section  are  parallel,  and  when- 
ever a  plane  intersects  an  object  whose  opposite  surfaces  are  parallel, 
the  opposite  sides  of  the  section  must  be  parallel ;  they  will  be  of 
equal  length  when  the  intersected  surfaces  are  equal  in  width  and 
the  intersecting  plane  cuts  the  entire  width  of  the  surfaces. 

155.  Sections  of  the  Cylinder.     A  section  of  a  cylinder  is  a 
circle  when  the   cutting  plane  is  perpendicular  to  the  axis  of  the 
cylinder,  a  rectangle  when  the  plane  is  parallel  to  the  ax's  of  the 
cylinder,  and  an  ellipse  when  it  is  oblique  to  the  axis  of  the  cylinder. 

Sections  A  and  B  require  no  further  explanations. 


156.    Section  C  is  an  ellipse  which  appears  a  straight  line  in  the 
front,  and  a  circle  in  the  top  view  ;  its  real  length  is  seen  in  the  front 


SECTIONS.  93 

view.  The  short  axis  of  the  ellipse  bisects  its  long  axis,  and  intersects 
the  axis  of  the  cylinder ;  it  appears  a  point  (IF  2^)  in  the  front  view, 
and  the  line  1-2  in  the  top  view. 

To  obtain  the  real  shape  of  the  section,  draw  ab  parallel  to 
aF  bF  and  set  off  upon  it  the  real  length  of  the  ellipse  by  means  of 
perpendiculars  to  aF  bF  from  the  points  aF,  bF ;  then  set  off  each 
side  of  o,  on  the  perpendicular  to  ab,  one  half  of  1—2  of  the  top  view. 
This  gives  the  short  axis  of  the  ellipse. 

To  obtain  other  points,  j>  and  4,  assume  a  point  (JF  4**)  in  aF  bF. 
This  represents  point  3  at  the  front  and  point  4  at  the  back  of  the 
section.  Project  from  3F  4F to  the  top  view  and  also  to  the  real 
shape  of  the  section,  where  the  distance  3—4.  is  to  be  made  equal  to 
3-4  of  the  top  view.  In  this  way  any  number  of  points  in  the  real 
shape  may  be  obtained. 

I57«  If  the  cylinder  is  placed  so  that  it  does  not  appear  a  circle 
in  any  view,  its  section  by  a  plane  may  be  obtained  by  assuming 
elements  upon  the  cylinder,  and  finding  the  points  in  which  they  are 
intersected  by  the  cutting  plane.  These  points  in  the  elements  are 
seen  in  the  view  in  which  the  cutting  plane  appears  a  line,  and  can 
be  projected  from  this  view  to  the  other  views. 

158.  Sections  of  a  Pyramid.  A  section  of  a  pyramid  is  a 
figure  similar  to  the  base,  when  the  cutting  plane  is  parallel  to  the 
base ;  a  triangle,  when  the  plane  passes  through  the  vertex  and  the 
base  of  the  pyramid,  or  when  it  intersects  the  base  and  two  of  the 
triangular  faces  of  the  pyramid.  When  the  pyramid  is  intersected  in 
any  other  way  the  section  will  be  a  polygon  having  a  side  upon  each 
face  of  the  pyramid  cut  by  the  plane. 

If  the  preceding  articles  are  understood,  these  statements  and 
drawings  A  and  B  will  be  clear  to  all. 

159.  In  C,  plane  AB  cuts  the  lateral  edges  of  the  pyramid  in 
four  points,  z,  2,  j,  4,  which  are  determined  in  the  front  view. 
Points  i  and  2  may  be  projected  from  this  view  to  the  top  view. 
Points  3  and  4  are  in  lateral  edges  which  are  represented  by  vertical 
lines  in  both  views,  and  therefore  these  points  cannot  be  obtained  in 
the  top  view  by  projecting  from  the  front  view. 

To  obtain  3  and  4  in  the  top  view  of  C,  it  is  necessary  to  take  an 
auxiliary  cutting  plane  through  3  and  4.  A  horizontal  plane  through 


94 


MECHANICAL   DRAWING. 


these  points  gives  a  square  section  whose  opposite  corners  aF  and 
bF  can  be  projected  to  the  top  view,  thus  giving  ar  and  br.  The 
corners  j  and  4  of  this  square  are  the  points  required  to  enable  the 
top  view  of  the  section  to  be  completed. 

160.  Sections  of  the  Cone.     Any  section  of  a  right  circular  cone 
made  by  a  plane  parallel  to   its  base  is  a  circle.     This  section  is 
illustrated  by  Fig.  88. 

161.  If  a  cutting  plane  passes  through  the  vertex  of  the  cone,  it 
intersects  the  base  of  the  cone  in  a  chord  of  the  circle,  and  the 
curved  surface  of  the  cone  in  two  elements ;  the  section  is  thus  a  tri- 
angle.    See  Fig.  96. 

162.  If  the  plane  intersects  all  the  elements  of  the  cone,  the  sec- 
tion is  an  ellipse.     Fig.  98. 

163.  If  a  cone  is  intersected  by  a  plane  which  is  parallel  to  one 
of  the  elements  of  the  cone,  the  section  is  a  parabola.     Fig.  99. 

164.  If  a  cone  is  intersected  by  a  plane  which  makes  a  greater 
angle  with  the  base  than  do  the  elements,  the  section  is  an  hyper- 
bola.    Fig.  97. 

165.  When  a  solid,  bounded  by  plane  surfaces,  is  cut  by  any 
plane,  the  edges  of  the  solid  are  seen  piercing  the  plane,  in  the  view 
in  which  the  cutting  plane  appears  a  line ;  these  points  of  intersec- 


SECTIONS. 


95 


tion  are  readily  projected  to  the  lines  of  the  other  views,  and  thus  the 
angles  of  the  section  are  obtained. 

166.  When  curved  bodies  or  those  with  curved  surfaces  are  in- 
tersected by  a  cutting  plane,  only  the  points  of  intersection  in  the 
contour  elements  are  seen  in  the  view  in  which  the  cutting  plane 
appears  a  line.  To  obtain  other  points  in  the  section,  it  is  necessary 
to  assume  elements  or  other  lines  upon  the  curved  surface ;  their 
intersections  will  be  found  in  the  same  way  as  the  intersections  of 
the  edges  of  solids  bounded  by  plane  surfaces.  When  possible  the 
lines  assumed  should  be  elements  of  the  surface.  Fig.  98. 

Instead  of  assuming  elements  in  order  to  obtain  the  points  in  a 
section,  it  is  often  easier  and  more  accurate  to  pass  through  the 
object  a  number  of  auxiliary  cutting  planes  whose  sections  are 
simple.  These  planes  intersect  the  plane  of  the  section  in  lines, 
whose  ends  must  be  points  in  the  required  section.  See  4  and  5, 
Fig.  97. 

If  pupils  can  obtain  the  sections  of  the  regular  geometric  solids 
they  will  readily  find  the  sections  of  any  solid  which  may  be  con- 
structed. 


CHAPTER    X. 

INTERSECTIONS. 

167.  IN  practical  work  it  is  necessary  to  represent  all  kinds  of 
regular  and  irregular  bodies  which  intersect  or  penetrate  each  other. 
The  knowledge  required  to  do  this  is  best  obtained  by  study  of  the 
geometric  solids. 

Simple  intersections  are  produced  when  a  body  small  enough 
to  pass  through  another  enters  one  plane  surface  and  leaves  by 
another.  The  large  object  being  bounded  by  plane  surfaces,  or 
at  least  by  the  two  mentioned,  the  intersections  are  simply  sections 
of  the  smaller  body  made  by  the  plane  surfaces  of  the  larger,  and 
have  been  explained. 

Simple  intersections  are  also  given  by  a  cone  or  cylinder  which 
penetrates  a  sphere  in  such  a  way  that  its  axis  passes  through  the 
centre  of  the  sphere.  In  this  case  the  plane  of  the  intersection 
must  be  at  right  angles  to  the  axis  of  the  penetrating  body  and  the 
lines  in  which  the  cylinder  or  cone  enters  and  leaves  the  sphere  must 
be  circles.  Fig.  100. 

If  the  axis  of  the  cone  or  cylinder  does  not  pass  through  the 
centre  of  the  sphere,  the  intersections  will  not  be  circles,  and  must  be 
obtained  as  explained  later. 

168.  The  principles  of  sections  enable  us  to  find  all  intersec- 
tions ;  for  if  the  lines  of  intersection  are  not,  in  part  or  in  whole, 
some  of  the  regular  sections  explained,  they  can  be  determined  by 
means  of   points   situated    in    sections   given   by  auxiliary  cutting 
planes. 

169.  When  bodies  bounded  by  plane  surfaces  intersect,  the  lines 
of  intersection  will  be  straight,  and  must  connect  the  points  in  which 
the  edges  of  each  solid  penetrate  the  other  solid  ;  the  problem  is  thus 
to  find  these  intersections. 

170.  When  curved  bodies  intersect  or  are  intersected,  there  are 
elements,  instead  of  edges,  which  penetrate  the  surfaces  and  must  be 
treated  as  if  they  were  edges.     Thus  problems  in  intersections,  as 


INTERS  EC  TIONS. 


97 


NOTE. 
assumed. 


they  are  generally  solved,  may  be  reduced  to  the  simple  problem  of 
finding  the  intersection  of  a  line  and  a  plane,  or  of  a  line  and  a 
curved  surface. 

INTERSECTIONS  OF  A  LINE  AND  A  PLANE  SURFACE. 

171.  Drawing  A  represents  a  cube  pierced  by  an  inclined  line 
which  enters  the  left  side  at  a  and  leaves  the  top  of 
the  cube  at  b.  The  front  view  determines  both  a 
and  l>,  for  it  represents  both  the  left  and  the  top  sur- 
faces by  straight  lines.  The  top  view  determines 
only  one  point  (a),  for  in  it  the  top  of  the  cube 
appears  a  surface,  and  when  a  plane  appears  a  surface 
its  intersection  by  a  line  cannot  be  determined  by  means 
of  this  view  alone.  Point  b  must  be  obtained  in  the 
top  view  by  projecting  from  the  front  view. 

The  positions  of  the  intersecting  lines  in  all  the  problems  are 

172.  Drawing  B  represents  a  cube  and  a 
line  i  which  is  in  the  same  plane  as  the  right 
and  left  vertical  edges  of  the  cube,  and  there- 
fore intersects  these  edges.  Line  2  is  in  front 
of  the  right  and  left  vertical  edges  ;  its  intersec- 
tions with  the  faces  of  the  cube  are  seen  in  the 
top  view,  and  may  be  projected  from  this  view 
to  the  front  view. 

173.  Drawing  C  represents  a  cube  intersected  by  a  line  which 
enters  the  left  visible  vertical  surface,  and  leaves 
at  the  top  of  the  cube.  The  front  view  represents 
the  top  of  the  cube  by  a  horizontal  line,  and  in  this 
view  the  intersection  bF  of  the  line  with  the  top  is 
seen. 

The  intersection  a  of  the  line  with  the  left  ver- 
tical surface  of  the  cube  is  seen  in  the  top  view. 
The  points  determined  by  each  view  can  be  pro- 
jected to  the  other  view,  and  in  this  way  each  view 
completes  the  other 


98 


MECHANICAL   DRAWING. 


I74«    Drawing  D  shows  a  line  which  pierces  the  left  end  and  the 

front  inclined  surface  of  a  triangular 
prism.  The  intersection  of  the  line 
and  the  inclined  surface  cannot  be 
determined  in  either  the  front  or  the 
top  view,  for  in  neither  view  does  the 
surface  appear  a  line.  In  the  side 
view  the  inclined  surface  appears  a 
line,  and  the  intersection  aL  is  seen  and 
may  be  projected  to  the  other  views. 

175.  Drawing  E  represents  a  square  pyramid  and  an  inclined 

line  L  which  pierces  the  pyramid  in  front  of  the  axis. 
The  intersections  of  the  line  and  the  lateral  faces  of 
the  pyramid  cannot  be  seen  in  either  view,  for  the 
faces  appear  surfaces  in  both ;  and  for  the  same 
reason  a  side  view  is  of  no  assistance.  To  find  the 
points,  an  auxiliary  cutting  plane  must  be  taken 
through  the  line.  If  a  vertical  plane  is  chosen  it 
will  appear  the  line  LT,  in  the  top  view,  and  in  this 
view  its  intersections,  i  and  2,  with  the  edges  of  the 
base  are  seen.  It  intersects  the  central  lateral  edge 
in  j,  which  is  found  as  explained  in  Art.  159.  In 
the  front  view  the  triangle  i,  2,  j  is  the  section  made  upon  the  pyra- 
mid by  the  vertical  cutting  plane  passing  through  Z.  The  points  a 
and  b,  where  L  intersects  the  sides  of  the  triangle,  are  the  points  in 
which  L  enters  and  leaves  the  surface  of  the  pyramid. 

176.  Drawing  F  represents  the  same  pyramid  and  line  as  draw- 

ing £,  but  instead  of  choosing  a  vertical  plane  through 
L  as  the  auxiliary  plane,  a  plane  is  passed  through 
Z,  perpendicular  to  the  front  plane.  The  front  view 
gives  the  points  IF,  2F,  JF,  and  4F  where  the  lateral 
edges  intersect  this  cutting  plane,  whose  figure  of 
intersection  is  readily  found  as  explained  in  Art.  159. 
In  the  top  view  this  section  appears  a  surface,  and 
the  points  in  it,  aT  and  1>T,  in  which  Z  enters  and 
leaves  the  pyramid,  are  seen  and  may  be  projected  to 
the  front  view. 


IN  TERSE  C  TIONS. 


99 


177-    Drawing  G  represents  the  same  conditions  as  D. 

The  intersection  may  be  found  without  drawing 
the  end  view,  by  means  of  a  cutting  plane  used  as 
explained  in  E  and  F. 

The  cutting  plane  used  in  the  illustration  is 
perpendicular  to  the  front  plane ;  and  in  the  front 
view  points  i,  2,  j  of  the  section,  which  are  in  the 
edges  of  the  prism,  are  seen.  When  these  points 
are  projected  to  the  top  view  and  connected,  the 
triangular  section  is  represented  by  a  triangle,  and 
the  point  aT,  where  LT  intersects  the  triangle,  is 
the  top  view  of  the  required  point  a. 


INTERSECTIONS  OF  A  LINE  AND  A  CURVED  SURFACE. 

178.  Drawing  //represents  a  sphere  intersected  by  a  horizontal 

line  Z,  in  front  of  the  centre  of  the  sphere. 

If  the  line  were  in  a  vertical  plane  passing 
through  the  centre  of  the  sphere  and  parallel  to 
the  front  plane,  the  front  view  would  give  the  in- 
tersections of  the  line  and  the  sphere.  If  the  line 
were  in  a  horizontal  plane  passing  through  the 
centre  of  the  sphere,  the  top  view  would  give  the 
intersections  of  the  line  and  the  sphere.  As  the 
line  is  not  situated  in  either  of  these  positions,  the 
intersections  cannot  be  in  the  circle  of  either  view, 
and  must  be  found  by  means  of  an  auxiliary  cut- 
ting plane. 

To  obtain  the  points  in  which  L  enters  and  leaves  the  sphere,  a 
horizontal  cutting  plane  may  be  passed  through  the  line.  This  plane 
gives  a  circular  section  whose  diameter  1-2  is  seen  in  the  front  view. 
The  circle  appears  a  circle  in  the  top  view;  here  its  intersections 
ar  and  bT  with  LT  are  seen,  and  may  be  projected  to  the  front 
view. 

179.  Drawing  /solves  the  same  problem,  except  that  the  line  is 
parallel  to  the  front  plane  only.     An  auxiliary  cutting  plane  parallel 
to  the  front  plane  is  used.     This  plane  gives  a  circle  on  the  sphere 


IOO 


MECHANICAL    DRAWING. 


whose  diameter,  1-2,  is  seen  in  the  top  view ;  its  real  shape  is  seen 
in  the  front  view,  where  a  and  b  are  determined, 
and  from  which  they  may  be  projected  to  the  top 
view. 

When  L  is  parallel  to  only  one  plane,  the  sec- 
tion must  be  taken  through  L  parallel  to  this  plane. 
If  L  is  not  parallel  to  either  plane  the  cutting  plane 
may  be  perpendicular  to  either  plane  and  will  give 
a  circle  which  in  one  view  will  appear  an  ellipse. 

180.  Drawing  J  represents  a  sphere  and  two 
inclined  lines  which  are  represented  in  the  top 
view  by  the  line  IT-2T,  and  in  the  front  view 
by  two  lines,  the  upper  of  which  intersects  the 
sphere  in  points  a  and  b  and  has  its  ends  un- 
marked ;  the  lower  line  intersects  the  sphere 
in  points  c  and  d,  and  its  ends  are  represented 
by  points  IF  and  2F.  The  upper  line  1-2 
intersects  the  sphere  at  points  a  and  />,  which 
are  upon  the  upper  surface  of  the  sphere, 
point  a  being  upon  the  front  surface.  Neither 
a  nor  b  appears,  in  either  view,  in  the  contour 
of  the  sphere,  and  to  determine  these  points, 
a  vertical  cutting  plane  is  taken  through  the 
line.  This  plane  intersects  the  sphere  in  a  circle,  which  in  the  front 
view  appears  an  ellipse.  Only  the  upper  half  of  this  ellipse  is 
drawn,  >as  it  contains  both  a  and  b.  (When  following  this  solution, 
cover  the  lower  half  of  the  front  view  with  paper,  to  hide  the  second 
solution  of  the  problem.) 

181.  If  the  lower  half  of  the  ellipse,  which  represents  the  section 
given  by  the  vertical  plane  through  the  lines,  is  drawn,  it  will  con- 
tain points  c  and  d,  in  which  the  lower  line  intersects  the  surface. 
These  points  may  also  be  found  by  revolving  the  vertical  plane  taken 
through  1-2,  until  it  is  parallel  to  the  front  plane.  The  section  given 
by  it  will  then  appear  a  circle  in  the  front  view.  As  the  plane 
revolves  about  a  vertical  axis  passing  through  the  centre  of  the 
circular  section,  the  points  of  the  section  and  points  i  and  2 
describe  arcs  of  circles,  which  appear  arcs  in  the  top  view,  and  hori- 


IN  TERSE  C  TIONS. 


101 


zontal  lines  in  the  front  view.     (The  upper  half  of  the  front  view  is 
to  be  covered  when  following  this  description.) 

As  the  vertical  cutting  plane  passing  through  line  1-2  revolves 
into  a  position  parallel  to  the  front  plane,  points  /  and  2  move,  in 
the  front  view,  in  horizontal  lines.  In  these  lines  points  i"  and  2" 
must  be  given  by  projecting  lines  from  i'  and  2'  which  represent  the 
ends  of  the  line  1—2  when  it  is  parallel  to  the  front  plane.  When  the 
vertical  plane  passed  through  1-2  has  been  revolved  parallel  to  the 
front  plane,  the  circle  which  it  gives  upon  the  sphere  will  appear  a 
circle,  whose  lower  half  is  shown  in  the  drawing ;  the  line  i"-2n  in- 
tersects this  circle  in  points  c'  and  d'.  These  points  move,  when  the 
plane  of  the  section  is  revolved  back  into  its  oViginal  position,  in  arcs 
which  appear  horizontal  lines  in  the  front  view,  and  which  intersect 
IF-2F  in  CF  and  dF  \  these  are  the  front  views  of  points  c  and  d,  in 
which  the  lower  line  1-2  intersects  the  surface  of  the  sphere.  These 
points  are  not  shown  in  the  top  view,  but  would  be  given  by  verticals 
from  CF  and  dF,  intersecting  ir—2r. 

182.  Drawing  K  represents  a  cylinder  intersected 
by  a  line  L. 

The  intersection  of  a  line  and  the  curved  surface 
of  a  cylinder  is  determined  in  the  view  in  which  the 
curved  surface  appears  a  circle.  If  the  curved  sur- 
face is  not  seen  edgewise  in  any  view,  the  cylinder 
must  be  cut  by  a  plane  passing  through  the  line,  in 
order  that  the  points  of  intersection  may  be  deter- 
mined ;  this  applies  to  any  and  all  positions  of  the 
cylinder  and  the  line. 

183.    Drawing  L  represents  a  cone,  and  a  line  which  intersects  it 
in  front  of  its  axis. 

If  the  line  were  in  the  plane  of  the  contour  elements, 
the  points  of  intersection  would  be  seen  in  the  front 
view,  but,  as  this  is  not  the  case,  to  obtain  the  points, 
an  auxiliary  plane  must  be  used.  A  horizontal  cutting 
plane,  A,  intersects  the  cone  in  a  horizontal  circle, 
which  appears  a  circle  in  the  top  view.  In  this  view 
the  intersections,  a  and  3,  of  the  line  A  and  the  circle 
are  determined,  and  may  be  projected  to  the  front 
view. 


K 


IO2 


MECHANJCAL   DRAWING. 


184.  Drawing  M  represents  a  cone  intersected  by  an  inclined 

line. 

To  obtain  the  points  of  intersection,  a  cutting  plane 
perpendicular  to  the  front  plane  may  be  used ;  this 
will  cut  the  cone  in  an  ellipse,  which  will  appear  an 
ellipse  in  the  top  view.  A  cutting  plane  perpendicular 
to  the  top  plane  is,  however,  used  here.  This  cuts  the 
cone  in  an  hyperbola,  whose  real  shape  is  seen  in  the 
front  view.  Art.  164.  The  intersections,  a  and  £,  of 
L  and  the  cone  are  seen  in  the  front  view,  and  from 
this  view  may  be  projected  to  the  top  view. 

185.  When  auxiliary  cutting  planes  are  used,  it  is  not  necessary 
to  find  the  complete  section  given  by  the  plane,  as  the  part  near  the 
point  of  intersection  is  all  that  is  required. 

INTERSECTIONS  OF  SOLIDS. 

186.  All   the  points    necessarily  involved   in   intersections   are 
explained  above.     If  understood  they  can  be  readily  applied  when, 
instead  of  a  single   line  penetrating  a  solid,  one  solid  penetrates 
another.     In  this  case,  when  the  points  of  intersection  of  the  various 
lines  of  the  penetrating  solid  cannot  be  seen  in  one  view  or  the 
other,  it  is  necessary  to  use  auxiliary  cutting  planes.     These  planes 
should  be  so  chosen  as  to  give  simple  sections  upon  both  solids. 
The  circle  and  the  rectangle  are  the  simple  sections  of  the  cylinder  ; 
the  triangle  and  the  circle  are  the  simple  sections  of  the  cone ;  and 
in  many  problems  it  will  be  possible  to  obtain  these  sections  instead 

of  ellipses,  hyperbolas,  and  parabolas. 

187.  When  a  cutting  plane  intersects  two 
intersecting  bodies,  it  gives  upon  each  a  cer- 
—  tain  figure.  These  figures  intersect  in  points, 
which  must  be  points  in  the  lines  in  which 
the  solids  intersect. 

This  is  illustrated  by  N  and  O.  In  N,  a 
vertical  cutting  plane  intersects  the  horizontal 
prism  in  a  rectangle,  and  the  sphere  in  a 
circle.  These  figures  intersect  in  four  points, 
which  are  points  in  the  lines  of  intersection 
of  the  prism  and  the  sphere. 


INTERSEC  TIONS. 


103 


In  O,  a  horizontal  cutting  plane  gives  a  rectangle  upon  the  cylin- 
der and  a  circle  upon  the  cone.  These  fig- 
ures intersect  in  four  points,  which  are  points 
in  the  lines  of  intersection  of  the  cylinder  and 
the  cone. 

The  same  problem  in  intersections  may  be 
solved  in  many  different  ways,  but  always  by 
means  of  auxiliary  cutting  planes,  when  the 
required  points  are  not  seen  at  once  in  one  of 
the  given  views.  These  planes  may  be  taken 
in  so  many  different  positions  that  to  explain 
all  would  be  impossible  in  the  limits  of  this 
book.  If  the  principles  of  the  subject  are 
understood,  pupils  will  have  no  trouble  in 
deciding  what  planes  will  give  the  simplest  sections,  or  in  determin- 
ing the  points  of  these  sections. 

Instead  of  using  parallel  cutting  planes,  as  explained  in  the  pre- 
ceding articles,  one  plane  may  be  supposed  to  be  hinged,  or  to  swing 
upon  a  given  line  as  an  axis.  In  the  case  of  two  intersecting  pyra- 
mids or  cones,  this  axis  should  pass  through  the  vertices  of  both 
solids,  thus  the  sections  of  both  will  be  triangles.  If  one  solid  is  a 
cylinder  or  prism  the  plane  should  swing  upon  a  line  parallel  to  the 
axis  of  this  solid.  This  method  is  not  illustrated,  but  is  often  of 
great  value. 


CHAPTER   XL 


ARRANGEMENT    AND    NAMES    OF    VIEWS. 

188.  IN    descriptive    geometry    the    planes    of    projection    are 
supposed  to  be  indefinite  in  extent,  one  horizontal  and  the  other 
vertical.      See    Fig.    35.      Four    dihedral    angles    are    formed    by 

these  planes.  The  eye  is  supposed  to  be  always 
in  front  of  the  vertical  plane  and  above  the  horizon- 
tal plane.  Objects  may  be  placed  in  any  of  these 
angles  and  projected  upon  the  two  planes ;  in  pro- 
jection and  descriptive  geometry  all  four  angles  are 
used. 

The  projection  of  an  object  upon  the  front  vertical  plane  is  called 
its  vertical  projection  ;  that  upon  the  horizontal  plane  is  called  its 
horizontal  projection.  These  drawings  are  also  called  elevation  and 
plan,  and  front  view  and  top  view.  All  these  terms  are  applied  in 
practical  work,  but  the  preference  seems  to  be  to  call  shop  drawings 
views.  In  the  study  of  projection  and  descriptive  geometry,  the 
drawings  are  generally  called  projections. 

189.  If  an  object  is  in  the  first  angle,  it  is  between  the  planes 
and  the  eye,  and  covers  its  projections  upon  the  planes. 


FIG.  35. 


FIG.  37. 


ARRANGEMENT  AND   NAMES   OF    VIEWS.  105 

Fig.  36  represents  the  vertical  and  horizontal  planes  forming  the 
first  angle,  with  a  side  vertical  plane  placed  at  the  right.  The  drawing 
also  shows  a  pyramid  placed  within  this  angle,  and  its  projections 
upon  the  three  planes. 

When  the  planes  are  revolved,  as  in  Fig.  37,  the  top  view  of  the 
pyramid  is  found  below  the  front  view,  and  the  view  of  the  left  side 
comes  at  the  right  of  the  front  view. 

This  arrangement  is  very  different  from  that  due  to  the  use  of  the 
third  angle,  which  is  explained  in  Chap.  Ill,  and  in  which  the  top 
view  is  above  the  front  view. 

190.  Draughtsmen    have  no   uniform  way  of  working.      Some 
arrange  the  views  according  to  the  use  of  the  first  angle,  and  some 
according  to  that  of  the  third  angle.     To  those  who  do  not  read 
drawings  easily  this  lack  of  uniformity  results   in  confusion.     To 
those  who  are   accustomed  to  reading  them,  the  positions  of   the 
views  are  of  less  importance,  and  the  convenience  of  the  draughts- 
man in  making  the  drawings  should  determine  largely  the  method  of 
arrangement. 

It  may  often  happen  that  it  is  not  easy  or  convenient  to  adhere 
to  any  system,  and  it  is  certainly  reasonable  that  a  rule  which  ham- 
pers should  not  be  followed.  Generally,  however,  some  uniform  ar- 
rangement of  views  may  easily  be  given,  and  most  draughtsmen  use 
either  that  given  by  the  first  or  that  given  by  the  third  angle.  The 
chief  points  to  decide  the  arrangement  to  be  adopted  should  be  the 
ease  and  accuracy  of  making  and  of  reading  the  drawing. 

191.  If  an  object  similar  to  that  illustrated  by  Fig.  38  be  placed 
in  the  first  angle,  the  draughtsman  will  be  obliged"  to  project  from 
the  right  end  of  the  front  view  across  its  entire  length  to  the  space  at 
the  left  of  the  front  view,  to  make  the  view  of  the  right  end  of  the 
object ;  and  in  the  same  way  across  the  entire  length  of  the  front  view 
to  give  the  view  of  the  left  end.     He  will  project  from  the  top  of  the 
front  view  to  the  space  below  this  view  to  draw  the  top  view ;  and 
from  the  bottom  of  the  front  view  to  the  space  above  this  view  to 
draw  the  bottom  view.     To  make  drawings  thus  arranged  must  take 
much  longer  than  to  make  them  arranged  as  in  Fig.  38,  with  the  view 
at  the  left  of  the  front  view  showing  the  left  end  of  the  object ;  that 
at  the  right,  the  right  end ;  that  below,  the  bottom  of  the  object,  and 


io6 


MECHANICAL    DRAWING. 


so  on.  Not  only  will  it  take  longer  to  make  the  drawings,  but  the 
inaccuracies  of  drawing  boards  and  T-squares  will  cause  the  drawings 
to  be  less  exact  than  those  whose  arrangement  is  due  to  the  use  of 
the  planes  placed  in  front  of  the  object. 

The  views  due  to  the  use  of  the  first  angle  are  not  so  easy  to  read 
as  those  due  to  the  use  of  the  third,  for  the  third  angle  places  the 
different  views  of  the  same  part  as  near  as  possible  to  each  other, 
while  the  first  angle  places  them  as  far  apart  as  possible.  These 
considerations  make  the  use  of  the  third  angle  desirable  for  all  prac- 


Front  View. 


Left  Side  View. 


Right  Side  View. 


tical  working  drawings  ;  and  therefore,  this  arrangement  is  adopted 
throughout  this  book. 

192.  In  the  study  of  projection  with  advanced  classes  the  choice 
of  angle  is  of  little  importance.  The  objects  may  be  placed  in 
either  of  the  angles,  and  there  will  be  little  difference  in  the  ease  (or 
perhaps,  difficulty)  with  which  the  students  understand  the  subject. 
A  point  in  favor  of  the  first  angle  for  use  in  the  study  of  projection 
is  that  books  on  the  subject  of  projection  generally  use  this  angle  ; 
another  point  is  that  when  the  third  angle  is  used,  and  glass  planes 
and  expensive  models  to  place  behind  them  are  not  provided,  the 


ARRANGEMENT  AND   NAMES   OF    VIEWS. 

problems  cannot  be  easily  illustrated.  When  the  first  angle  is  used 
two  blackboards  may  be  hinged  together,  as  illustrated  in  Fig.  36  ; 
they  should  be  arranged  so  that  they  may  revolve  while  fixed  at  right 
angles  to  each  other,  or  revolve  independently  so  as  to  form  one 
vertical  surface.  Thus  the  students  may  see  the  horizontal  board 
edgewise  and  the  vertical  one  as  a  surface  ;  or  the  horizontal  one  as 
a  surface  and  the  vertical  one  edgewise  ;  or  with  the  plane  of  the 
horizontal  one  coinciding  with  that  of  the  vertical,  as  in  Fig.  37.  A 
vertical  blackboard  may  be  attached  at  either  side  of  the  vertical 
board,  as  illustrated  in  Fig.  36,  to  represent  a  side  vertical  plane. 
By  means  of  these  boards  and  of  models  which  can  be  procured 
with  little  expense  and  held  as  desired  in  front  of  the  planes,  the 
problems  may  be  much  more  easily  illustrated  than  when  the  third 
angle  is  used.  These  boards  are  readily  managed  and  can  be  seen  by 
all  the  occupants  of  a  large  class-room,  so  that  for  advanced  classes 
who  are  studying  the  theory  of  projection,  the  first  angle  seems 
preferable,  especially  when  the  course  is  preparatory  to  one  in 
descriptive  geometry.  But  although  books  on  this  subject  use 
principally  the  first  angle,  all  are  necessary,  and  a  thorough  course 
will  deal  with  points,  lines,  planes,  and  objects  in  all  the  angles. 

The  third  angle  as  well  as  the  first  might  be  used  principally  in 
the  study  of  descriptive  geometry.  The  fact  that  the  arrangement  of 
views  given  by  the  first  angle  is  different  from  that  which  we  have 
decided  to  be  the  best  for  shop  drawings  should  not,  however,  be  of 
the  slightest  consequence  to  the  advanced  student,  for  he  should 
study  the  relations  of  points  in  all  the  angles ;  and  he  should  be  as 
familiar  with  the  arrangement  due  to  the  third  angle  as  he  is  with 
that  due  to  the  first.  When  his  course  is  completed  and  he  makes 
practical  drawings,  he  will  then  make  them  with  equal  ease,  whatever 
arrangement  of  views  he  may  choose. 

Projection  and  descriptive  geometry  are  means  by  which  the 
mind  is  trained  to  conceive  any  form  in  any  position ;  their  aim  is 
not  the  simpler  and  less  important  one  of  enabling  young  students  to 
make  views  or  working  drawings  of  concrete  objects. 

The  question  as  to  whether  to  use  solids,  planes,  lines,  or  points 
for  the  first  study,  has  caused  some  discussion.  From  what  has 
been  said,  it  will  be  seen  that  for  the?  first  work  in  the  public  schools 


IO8  MECHANICAL   DRAWING. 

and  for  elementary  work  generally,  it  will  be  better  to  begin  with 
solids  ;  for  to  deal  with  lines,  etc.,  is  more  in  accordance  with  pro- 
jection than  with  the  making  of  views  from  simple  objects. 

In  the  theoretical  study  of  projection  it  makes  no  difference  what 
is  taken  first ;  for  if  the  solid  is  taken  the  lesson  still  must  deal  with 
the  relations  of  points  and  lines  to  each  other  and  to  the  planes  of 
projection,  so  that  in  reality  these  are  studied  first. 


CHAPTER   XII. 

PLATES   AND    EXPLANATIONS. 

THE  plates  of  this  chapter  illustrate  work  from  that  suited  for  the 
youngest  pupils  of  the  subject,  to  that  which  advanced  pupils  of  high 
and  elementary  technical  schools  may  require.  Teachers  are  to 
select  work  suitable  for  their  pupils  and  make  it  as  practical  as 
possible  by  following  the  directions  given  in  the  preceding  chapters. 

The  principles  of  working  drawings  may  be  taught  by  means  of 
free-hand  sketches.  Free-hand  sketches  may  also  be  made  and 
dimensioned  in  order  that  finished  drawings  may  be  made  from 
them.  All  other  drawings  should  be  made  by  the  use  of  instruments, 
and  to  some  fixed  scale.  In  the  lower  grades  of  the  public  schools 
an  architect's  scale  cannot  be  provided,  and,  when  the  objects 
studied  cannot  be  represented  full  or  half  size,  a  special  scale  must 
be  drawn  as  explained  in  Art.  10. 

The  drawings  of  the  plates  are  small,  and,  in  order  that  they  may 
not  be  obscured,  working  or  projecting  lines  are  given  only  when 
necessary  to  illustrate  important  principles  or  ways  of  working ;  and 
dimensions  are  given  only  in  a  few  cases,  but  sufficient  to  show  clearly 
how  dimensions  should  be  placed.  In  some  of  the  drawings,  dimen- 
sion lines  and  arrow-heads  are  placed  where  dimensions  should  be 
given  ;  this  is  done  so  that  the  drawings  may  not  be  used  for  copies, 
and  that  objects  similar  to  those  illustrated  may  be  studied,  measured, 
and  drawn  to  scale. 

The  models  represented  are  often  different  in  proportion  from 
the  regular  drawing  models ;  the  proportions  are  chosen  to  present 
the  principles  by  drawings  as  large  as  the  plates  will  allow. 

Those  who  understand  the  preceding  chapters  will  require  no 
further  explanation  of  the  points  there  considered ;  therefore  explana- 
tions of  such  points,  if  given  at  all,  will  be  stated  very  briefly,  and 
often  simply  by  reference  to  preceding  articles. 


I  I O  MECHANICAL   DRA  WING. 

The  plates  are  reproduced  from  drawings  twice  their  size.  The' 
lines  of  the  original  drawings  are  suitable  in  width  for  practical 
drawings.  The  lines  of  the  plates  are  finer  than  is  required  for 
any  except  the  most  highly  finished  drawings  made  by  advanced 
pupils. 

The  lines  of  the  drawings  upon  pages  61  and  106  are  suitable 
in  width  for  practical  drawings  and  for  pupils'  work. 


112  MECHANICAL   DRA  WING. 


PLATE    I. 

FIG.  39.     Front  and  top  views  of  a  sphere. 

Any  view  of  a  sphere  must  be  a  circle,  whose  diameter  is  equal  to  that 
of  the  sphere.  The  centres  of  these  views  are  in  a  vertical  line. 

FIG.  40.  Front  and  top  views  of  an  hemisphere  whose  plane  surface 
is  horizontal  and  uppermost. 

The  horizontal  circle  is  represented  by  a  circle  in  the  top  view,  and  by 
a  horizontal  line  in  the  front  view  (Art.  134)  ;  the  curved  surface  of  the 
hemisphere  is  represented  in  the  front  view  by  a  semi-circle  extending  from 
the  horizontal  line  downward. 

FIG.  41.  Front  and  top  views  of  a  cube,  two  faces  being  horizontal, 
and  one  face  appearing  a  square  in  the  front  view. 

The  top  face  appears  a  square  in  the  top  view. 

For  the  development  and  that  of  any  simple  solid  in  the  following  fig- 
ures, see  Chap.  IV. 

FIG.  42.  Front  and  top  views  and  development  of  a  right  square 
prism. 

FIG.  43.  Front,  top,  and  right  side  views  of  a  vertical  square  tablet 
so  placed  that  the  front  view  gives  its  real  shape. 

See  Art.  133. 

FIG.  44.  Front  and  top  views  of  a  vertical  oblong  tablet,  which  ap- 
pears its  real  shape  in  the  front  view. 

See  Art.  j  33. 

FIG.  45.  Front  and  left  side  views,  and  development  of  a  horizontal 
cylinder,  whose  axis  is  parallel  to  the  front  plane. 

The  front  view  is  a  rectangle  and  represents  the  circles  by  vertical  lines ; 
the  distance  between  these  verticals  is  equal  to  the  length  of  the  cylinder. 
The  side  view  is  a  circle,  and  should  be  drawn  first. 

The  form  of  the  laps  by  which  the  parts  are  fastened  together  is  imma- 
terial ;  that  shown  in  the  figure  may,  in  the  case  of  the  cylinder  or  cone, 
give  the  most  satisfactory  results. 


PLATE  I 


F.g.  39. 


Fig.  40. 


Fig-  41, 


Fig.  42, 


Fig.  44. 


Fig  43 


Fig.  45., 


MECHANICAL   DRAWING. 


PLATE    II. 

FIG.  46.  Front  and  left  side  views,  and  development  of  an  equilateral 
triangular  prism,  placed  horizontally  and  so  that  the  side  view  is  a  tri- 
angle. 

The  side  view  gives  the  real  shape  of  the  triangle  and  should  be  drawn 
first  (Art  135). 

The  front  view  is  an  oblong,  whose  length  is  equal  to  that  of  the  prism, 
and  whose  height,  given  by  projecting  from  the  side  view,  is  equal  to  1-2  of 
that  view  (Art.  138). 

FIG.  47.  Front  and  top  views  of  a  circular  tablet  which  appears  its 
real  shape  in  the  front  view., 

See  Art.  133. 

FIG.  48.  Front  and  top  views  of  an  equilateral  triangular  tablet 
which  appears  its  real  shape  in  the  front  view. 

See  Art.  133. 

FIG.  49.  Front  and  top  views  and  development  of  an  upright  square 
pyramid,  placed  so  that  the  edges  of  its  base  are  at  45°  to  the  front  plane. 

The  top  view  is  a  square  and  should  be  drawn  first  (Art.  134).  The 
lateral  edges  are  represented  in  this  view  by  the  diagonals  of  the  square. 
In  the  front  view  the  length  of  the  axis  is  seen,  also  the  real  length  of  the 
right  and  left  lateral  edges  ;  the  nearest  lateral  edge  appears  a  vertical  line 
as  long  as  the  axis,  and  thus  shorter  than  its  actual  length  (Art.  130). 

FIG.  50.     Front  and  top  views  of  a  vertical  tumbler. 

At  the  top  and  bottom  of  the  object  are  horizontal  circles,  which  appear 
concentric  in  the  top  view.  Draw  the  top  view  of  the  outside  upper  edge  of 
the  tumbler  ;  then  the  front  view  of  the  tumbler  ;  and  from  the  dotted  line, 
showing  the  thickness  of  the  glass,  obtain  the  dimensions  of  the  inside 
circles  of  the  top  view. 

FIG.  5 i  •  Front  and  top  views  of  a  tin  cookie-cutter,  placed  so  that  in 
the  front  view  the  handle  is  a  semi-circle. 

The  top  view  of  the  cutter  is  a  circle  and  should  be  drawn  first ;  then 
draw  the  front  view  of  the  cutter  and  add  the  handle ;  draw  last  the  handle 
in  the  top  view. 

FIG.  52.      Top  and  front  views  of  a  cylindrical  box. 

The  thickness  of  the  top  and  bottom  is  shown  by  the  dotted  lines. 

Draw  the  top  view  first  (Art.  134). 

FIG.  53.     Front  and  top  view's  of  a  tin  dipper. 

Draw  the  top  view  of  the  dipper  first ;  then  the  front  view  ;  then  the 
front  view  of  the  handle  ;  and  lastly  the  top  view  of  the  handle. 


PLATE 


Fig.  46. 


Fig.  50. 


Fig.  51. 


Fig.  49. 


Fig.  47. 


Fig.  48. 


Fig.  52. 


Fig.  53. 


Il6  MECHANICAL   DRAWING 


PLATE    III. 

FIG.  54.  Top  and  front  -views  of  an  upright  hexagonal  prism,  and 
the  development  of  the  same. 

FIG.  55.  Top  and  front  views  and  development  of  an  upright  hexag- 
onal Pyramid. 

FIG.  56.     Front  and  top  views  and  development  of  an  upright  cone. 

FIG.  57.     Front,  top,  and  right  side  views  of  an  hexagonal  tablet. 

FIG.  58.  Front  and  top  views  of  a  tin  tunnel,  placed  so  that  its  circu- 
lar edges  appear  circles  in  the  top  view. 

FIG.  59.     Top  and  front  views  of  a  tin  grater. 

In  all  the  figures  of  this  plate,  except  Fig.  57,  the  top  view  should  be 
drawn  first.  In  Fig.  57  the  front  view  should  be  drawn  first. 


PLATE  III 


Il8  MECHANICAL   DRAWING. 


PLATE    IV. 

FIG.  60.     Front,  top,  and  right  side  views  of  a  knife-box. 

FIG.  61.      Top  and  front  views  of  an  oil-can. 

Draw  the  top  view  first. 

FIG.  62.      Top  and  front  views  of  a  call-bell. 

FIG.  63.  Front  and  left  side  views  of  a  horizontal  hollow  cylinder;  a 
horizontal  section  on  AB,  and  a  cross  section  of  the  cylinder  on  CD. 

Draw  the  side  view  or  the  cross  section  first. 

FIG.  64.     Top  and  front  views  of  a  dish  with  a  handle. 

Draw  the  top  view  of  the  dish  first ;  then  the  front  view  ;  then  draw  the 
handle. 

FIG.  65.     Front  and  top  views  of  a  mallet. 

Draw  the  front  view  of  the  head  first,  and  then  carry  the  drawing  of  the 
two  views  along  together,  as  explained  in  Art.  36. 

All  the  above  drawings  may  be  made  most  advantageously  by  carrying 
all  the  views  of  each  object  along  at  the  same  time,  as  explained  in  Art.  36. 
It  is,  however,  not  necessary  or  important  that  young  pupils  should  work  in 
this  way.  With  them  the  question  is  not  speed,  but  accuracy  and  under- 
standing of  the  principles :  hence .  they  may  begin  with  the  view  which  is 
easiest  to  draw,  and  may  carry  it  as  far  as  possible,  and  even  finish  it, 
before  beginning  any  other  view. 


PLATE  IV. 


I-    M 


Fig.  60. 


_J 

SECTION  AT  A  B 


— •   -t— 


Fig.  63. 


c  i 


' 

SECTION  AT  C  D 


\ 


;    Fig.  64. 


Fig.  65 


I2O  MECHANICAL   DRAWING. 


PLATE    V. 

FIG.  66.      Top  and  front  views  of  a  tin  coffee-pot. 

Draw  the  top  view  of  the  pot  first ;  then  the  front  view,  adding  the 
handle  and  nose,  first  in  this  view,  and  then  in  the  top  view. 

FIG.  67.  Front',  and  left  side  views,  and  a  longitudinal  section  of  a 
tool  handle. 

Draw  the  side  (end)  view  first,  and  the  other  two  views  as  explained 
in  Art.  36.  The  handle,  being  of  wood,  may  be  represented,  if  desired,  as 
shown  on  page  61. 

FIG.  68.     Front,  top,  and  left  side  views  of  aflatiron. 

FIG.  69.  A  front  view,  a  section  on  CD,  and  a  horizontal  section  at 
AB  through  a  horizontal  spool. 

Draw  the  section  on  CD  first. 

FIG.  70.  Front  and  top  views  of  a  vertical  circular  tablet,  attached  to 
the  back  edge  of  a  horizontal  square  tablet. 

FIG.  71.  Front  and  top  views  of  a  vertical  hexagonal  tablet,  attached 
to  the  back  edge  of  a  horizontal  square  tablet. 


PLATE  V. 


122  MECHANICAL   DRAWING. 


PLATE    VI. 

FIG.  72.  Top,  front,  and  right  side  views  of  a  horizontal  oblong 
tablet,  with  a  vertical  triangular  one  attached  to  its  right  edge. 

Draw  the  top  or  side  view  first. 

FIG.  73.  The  same  views  of  the  same  combination  as  in  Fig.  72,  but 
with  a  vertical  pentagonal  tablet  attached  at  the  back  edge  of  the  horizontal 
tablet. 

Represent  the  pentagonal  tablet  in  the  front  view  first. 

FIG.  74.  Front,  top,  and  left  side  views  of  a  chair,  represented  by  com- 
bining a  horizontal  square  tablet  for  the  seat  with  vertical  square  tablets 
for  the  front  and  back  of  the  lower  part,  and  a  vertical  square  tablet  for 
the  back  of  the  chair. 

Draw  the  top  view  first. 

FIG.  75.  Front,  top,  and  right  side  views  of  a  square  prism,  support- 
ing an  oblong  tablet  at  an  angle  of  43°. 

Draw  the  side  view  of  the  prism  first ;  then  the  front  and  top  views. 
Draw  the  front  view  of  the  tablet  first ;  then  the  side  and  top  views. 

FIG.  76.  Front,  top,  and  right  side  views  of  a  circular  plinth,  or  of 
tablets  arranged  in  the  form  of  a  plinth,  supporting  a  square  plinth,  or 
tablets  arranged  in  the  form  of  a  plinth. 

Draw  the  top  view  of  the  circular  plinth  first ;  then  the  front  and  side 
views.  Draw  the  front  view  of  the  square  plinth  first,  and  then  the  top  and 
side  views. 

FIG.  77.  Top  and  front  views  of  a  square  plinth  with  a  circular 
plinth  resting  upon  it. 

Draw  the  top  view  first. 


Fife.  72. 


Fig.  73 


Fig.  74. 


Fig.  75. 


Fig.  76. 


Fig.  77. 


124  MECHANICAL  DRAWING. 


PLATE   VII.  , 

FIG.  78.  Top,  front,  and  right  side  views  of  an  hexagonal  plinth  with 
a  circular  plinth  upon  it. 

Draw  the  top  view  first.  Make  the  width  of  the  side  view  equal  to  1-2 
of  the  top  view. 

FIG.  79.  Top,  front,  and  right  side  views  of  a  circular  plinth  with  a 
square  plinth  upon  it. 

Draw  the  top  view  nrst. 

FIG.  80.  Front,  top,  and  right  side  views  of  a  triangular  prism  sup- 
porting an  oblong  tablet  at  an  angle  of  4.3°. 

Draw  the  side  view  first 

FIG.  81.  (A)  Front  and  toP  views  of  an  hexagonal  tablet,  when  it  is 
parallel  to  the  front  plane. 

(.Z?)  Front  and  top  views  of  the  same  hexagonal  tablet  when  it  is  at  an 
angle  of  60°  to  the  front  plane  and  perpendicular  to  the  top  plane,  as  in  A. 

Draw  the  front  view  of  A  first ;  then  the  top  view.  Place  the  top  view 
at  60°  and  project  from  it  and  from  the  front  view  of  A,  to  complete  B,  as 
explained  in  Art.  109. 

FIG.  82.  (A~)  Front  and  top  views  of  a  square  prism,  two  of  its  ob- 
long faces  being  at  45°  to  the  tofi  plane  and  the  other  two  being  parallel  to 
the  front  plane. 

(B}  Front  and  top  views  of  the  same  square  prism  when  two  of  its 
oblong  faces  are  at  45°  with  the  top  plane,  and  the  other  two  are  at  30° 
to  the  front  plane. 

Draw  the  front  view  of  A  first :  then  the  top  view.  For  the  top  view  of 
B  place  the  top  view  of  A  at  30°  and  then  obtain  the  front  view  of  B,  as 
explained  in  Arts.  112  and  113. 

At  C  are  given  front  and  too  views  of  the  prism  when  in  an  upright 
position. 

The  exponents  F,  T.  and  S  used  in  this  and  following  figures  indicate 
the  different  views,  —  front,  top.  and  side.  To  avoid  confusion  they  are  not 
placed  upon  all  the  figures  and  some  of  the  points  are  not  numbered  or 
lettered. 


PLATE  VII. 


Fig.  78. 


Fig.  79. 


Fig.  80. 


Fig.  81. 


Fig.  82. 


126  MECHANICAL   DRAWING. 


PLATE   VIII. 

FlG.  83.  Front,  top,  and  right  side  mews  of  the  lower  portion  of  an 
upright  square  prism,  cut  by  a  plane  at  an  angle  of  43°  to  its  base  ;  also 
the  real  shape  of  the  section  and  the  development  of  the  surface  of  the  object. 

Draw  the  top  view  first,  next  thje  front  and  side  views,  then,  in  the  front 
view,  the  line  of  the  cutting  plane  at  45°.  Project  the  section  to  the  side 
view  and  obtain  its  real  shape  as  explained  in  Art.  152. 

Develop  the  lateral  surface  of  the  prism  as  explained  in  Art  54.  To 
obtain  the  line  of  intersection  upon  it,  measure  (in  the  front  or  side  view) 
the  distances  of  points  /,  2, 3,  and  ^  from  the  base  of  the  object  and  set  these 
distances  off  on  the  development,  obtaining  /#,  2°,  JD,  and  4°.  Joining 
these  points  in  the  lines  defining  each  lateral  surface  gives  the  line  of  inter- 
section. Place  the  bases  upon  any  desired  lines  of  the  development  of  the 
bases. 

FIG.  84.  Top,  front,  and  right  side  views  of  an  upright  square  prism 
whose  lateral  faces  are  at  ^5°  to  the  front  plane,  and  which  is  intersected 
by  a  plane  at  an  angle  with  its  base ;  also  the  real  shape  of  the  section  and 
the  development  of  the  lateral  surface  of  the  object. 

See  Arts.  152  and  54. 

FIG.  85.  Front,  top,  and  right  side  views  of  a  horizontal  triangular 
prism;  a  section  of  the  same  by  a  plane  at  60°  to  its  horizontal  face;  also 
the  development  of  its  lateral  faces . 

Draw  the  side  view  first,  then  the  front  and  top  views,  then  the  line  of 
the  section  at  60°  in  the  front  view,  then  the  top  view  of  the  section,  and 
last  the  real  shape  of  the  section.  The  lengths  A-2D,  B-JD,  etc.,  for  the 
development  are  seen  in  the  front  and  top  views.  The  distance  between  A 
and  B,  etc.,  is  seen  in  the  side  view. 


PLATE  VIII. 


Fig.  84. 


Fig.  85. 


128  MECHANICAL   DRAWING. 


PLATE    IX. 

FIG.  86.  Top,  front,  and  right  side  views  of  a  vertical  hexagonal 
prism  cut  by  a  plane  at  4^  to  its  basej  also  the  real  shape  of  the  section 
and  the  development  of  the  surface. 

FIG.  87.  Top  and  front  views  of  the  frustum  of  a  square  pyramid, 
and  the  development  of  its  surface. 

Draw  the  top  view  of  the  complete  pyramid  first ;  next  its-  front  view  ; 
and  then  the  section  in  the  front  view;  project  from  the  front  view  the 
points  for  the  square  which  is  the  top  view  of  the  section  and  shows  its  real 
shape. 

The  length  of  the  lateral  edges  of  the  entire  object  is  seen  in  the  front 
view  from  IF  to  2F,  and  the  length  of  the  part  below  the  cutting  plane  is  seen 
from  2F  to  3F.  The  sides  of  the  base  appear  their  real  length  in  the  top  view. 

See  Art.  59. 

FIG.  88.     Top  and  front  views  and  development  of  the  frustum  of  a 
cone. 

To  develop,  divide  one  quarter  of  the  base  into  three  equal  parts. 
Draw  an  arc  with  radius  IF-2F.  Upon  this  arc  set  off  2T~4T  twelve  times. 
The  development  of  the  line  of  the  section  may  be  obtained  by  considering 
the  section  as  the  base  of  a  cone  extending  from  the  vertex  to  the  cutting 
plane.  The  elements  of  this  cone  are  equal  in  length  to  the  distance  IF-3F. 

FIG.  89.  Top,  front,  and  right  side  views  and  development  of  an  up- 
right square  pyramid,  intersected  by  a  plane  at  4^  to  its  base. 

The  lateral  edges  of  the  pyramid  are  not  parallel  to  the  front  or  side 
planes,  and  do  not  appear  their  real  lengths  in  the  front  or  side  view.  To 
develop  the  surface,  the  length  of  the  lateral  edges  must  be  found  by  revolv- 
ing one  of  them,  as  1-2,  until  in  the  top  view  it  is  horizontal  and  at  i'r-2 '. 
As  the  pyramid  revolves  upon  its  axis  until  the  lateral  edge  comes  to  this 
position,  point  2  moves  in  an  arc  which  appears  an  arc  in  the  top  view  and  a 
horizontal  line  in  the  front  view,  and  2"  will  be  under  21,  and  on  the  level  of 
2p,  and  the  distance  Z"-IF  must  be  the  real  length  of  the  edge  1-2.  Point  3, 
in  r-2,  also  describes  an  arc  in  the  top  view,  but  it  is  not  necessary  to  draw 
the  arc,  as  3"  must  be  in  iF-2"  and  on  the  level  of  3F.  The  distance  iF-3" 
is  the  real  distance  from  the  vertex  to  the  points  of  the  section  which  are 
in  the  two  left  lateral  edges.  The  real  length  of  the  distance,  1-4,  from  the 
vertex  to  the  points  in  the  two  right  lateral  edges  is  found  in  a  similar  way. 


PLATE  IX. 


130  MECHANICAL   DRAWING. 

PLATE    X. 

FIG.  90.  (A)  Front  and  top  -views  of  a  vertical  circular  card,  par- 
allel to  the  front  plane. 

(B)  The  same  views  of  the  same  card  when  perpendicular  to  the  top 
plane  and  at  an  angle  of 45°  with  the  front  plane. 

The  views  of  the  first  position  are  necessary  to  obtain  those  of  the  second, 
and  the  top  view  of  A,  when  it  is  placed  at  45°,  will  be  the  top  view  of  B. 

Draw  the  horizontal  and  vertical  diameters  in  the  front  view  of  A  and 
mark  their  ends  i,  3  and  2,  4.  Place  these  points  in  both  the  top  views 
and  obtain  them  in  the  front  view  of  B  by  verticals  from  the  top  view  of  B 
intersected  by  horizontals  from  the  front  view  of  A.  These  points  are  the 
ends  of  the  diameters  of  the  ellipse,  which  is  the  front  view  of  B.  To  find 
other  points,  place  any  point  as  CF  in  the  front  view  of  A.  Then  place  CT  in 
both  top  views,  and  find  C  in  the  front  view  of  B  as  explained.  If  more 
than  four  points  are  desired  it  is  best  to  take  eight  or  twelve  equidistant  points. 

FIG.  91.  (A}  Front  and  top  views  of  a  cone  resting  upon  an  element 
which  is  horizontal  and  parallel  to  the  front  plane. 

(jff)  Top  and  front  views  of  the  same  cone,  when  the  element  upon 
which  it  rests  is  horizontal  and  at  an  angle  of jo°  to  the  front  plane. 

The  front  view  of  A  will  be  an  isosceles  triangle  of  which  the  lower  of  the 
two  equal  sides  is  horizontal ;  the  base  of  the  triangle  represents  the  base  of 
the  cone.  If  the  dimensions  of  the  base  and  axis  of  the  cone  are  given,  the 
length  of  the  elements  must  be  found  by  drawing  an  isosceles  triangle 
whose  base  is  equal  to  the  diameter  of  the  base  of  the  cone  and  whose  alti- 
tude is  equal  to  the  axis  of  the  cone. 

To  obtain  the  front  view  of  A,  draw  the  element  /-p  horizontal  and  its 
real  length.  With  gF  as  a  centre  and  gF-iF  as  radius,  describe  an  arc 
through  point  IF.  With  IF  as  a  centre  and  the  diameter  of  the  base  of  the 
cone  as  a  radius,  describe  an  arc  to  intersect  the  arc  from  QF  in  point  5F. 
Join  IF-5F  and  5F-QF ;  this  gives  the  front  view  of  the  cone.  Bisect  the 
line  IF-JF  and  obtain  the  centre  of  the  base  of  the  cone. 

The  top  view  of  the  base  is  an  ellipse  which  may  be  found  as  explained 
in  Fig.  90.  To  obtain  the  points  by  which  the  ellipse  may  be  determined, 
the  circle  of  the  base  must  be  revolved  until  it  is  parallel  to  the  front  plane  ; 
it  may  then  be  divided  as  desired  and  the  points  revolved  back  to  the  line 
IF-5F.  As  the  circle  revolves  the  points  in  it  move  in  lines  which  in  the 
front  view  appear  parallel  to  the  axis  of  the  cone. 

In  the  top  view  the  circle  must  be  drawn  when  revolved  until  parallel  to 
the  top  plane ;  the  points  marked  in  the  circle  of  the  front  view  can  then  be 
placed  in  that  of  the  top  view.  As  the  circle  revolves  these  points  move,  in 


132 


MECHANICAL   DRAWING. 


this  view,  in  lines  parallel  to  the  axis  of  the  cone,  and  these  lines  intersected 
by  verticals  from  the  points  of  the  front  view  give  the  points  in  the  ellipse. 

The  top  view  of  B  is  the  top  view  of  A  with  its  axis  at  an  angle  of  30°. 
Having  drawn  this  top  view  the  points  of  the  front  view  may  be  found  as 
explained  in  Art.  1 1 2. 

FIG.  92.  Front  and  top  views  of  a  cylinder  whose  axis  is  at  43°  to 
the  top  plane  and  parallel  to  the  front  plane. 

Draw  the  front  view  first.  Obtain  the  top  view  of  the  circles  by  means 
of  points  in  the  circles,  as  in  Fig.  91. 

To  place  more  than  four  points,  draw  a  circle  upon  1-2  of  the  front 
view  and  upon  3-4.  of  the  top  view ;  divide  these  circles  as  desired,  and  so 
number  the  divisions  that  any  point  in  the  base  is  represented  in  both  circles 
by  the  same  figure  or  letter. 

Project  the  points  in  the  circle  drawn  in  the  front  view  to  line  IF-2F, 
and  from  this  line  project  the  points  to  the  top  view,  where  they  will  be 
situated  in  horizontal  lines  through  the  points  of  the  circle  there  drawn. 
This  amounts  to  revolving  the  circular  base  so  that  it  is  parallel  first  to  the 
front  plane  and  then  to  the  top  plane,  as  explained  in  Fig.  91. 

FIG.  93.  Front,  top,  and  right  side  views  of  a  square  pyramid,  whose 
azis  is  parallel  to  the  front  plane  and  at  4^  to  the  top  plane.  The  edges 
of  the  base  are  at  45°  to  the  front  plane. 

The  edges  of  the  base,  being  at  angles  to  both  the  front  and  the  top 
planes,  do  not  appear  their  real  length  in  either  view.  One  diagonal  of  the 
base  is  parallel  to  the  front  plane,  the  other  diagonal  is  parallel  to  the  top 
plane  ;  each  diagonal  appears  its  real  length  upon  the  plane  to  which  it  is 
parallel,  and  thus  the  base  of  the  pyramid,  which  is  perpendicular  to  the 
front  plane,  is  represented  in  the  front  view  by  a  line  whose  length  is  equal 
to  the  diagonal  of  the  base.  If  the  length  of  the  edge  of  the  base  is  given, 
that  of  the  diagonal  must  be  found  by  drawing  the  square,  or  one  half  of  it, 
as  at  A.  The  front  view  should  be  drawn  first,  then  the  top  or  side  view. 

FIG.  94".  (A)  Front,  top,  and  right  side  views  of  a  square  plinth  sup- 
porting a  triangular  prism. 

(2?)  Top  and  front  views  of  the  same  objects  when  the  vertical  faces  of 
the  plinth  are  at  JO°  and  60°  with  the  front  plane. 

(A)  First  draw  the  views  of  the  plinth ;  then  draw  the  prism  in  the 
front  view.  The  distance  1-2  of  this  view  is  equal  to  the  altitude  of  the 
triangular  end  of  the  prism.  To  obtain  1-2,  supposing  that  the  length  of 
the  side  of  the  triangle  is  given,  draw  the  real  shape  of  the  triangle  as  at  C. 
From  the  front  view  of  the  prism  obtain  the  other  views. 

(2?)  Place  the  top  view  of  A  at  the  required  angle,  and  obtain  the  front 
view,  as  explained  in  Arts.  112  and  113. 


PLATE  X. 


Fig.  92. 


Fig.  91. 


134  MECHANICAL   DRAWING. 


PLATE   XI. 

FlG.  95.  Top,  front,  and  right  side  views  of  an  hexagonal  Pyramid 
intersected  by  a  plane  oblique  to  the  base;  the  true  shape  of  the  section 
made  by  the  plane,  and  the  development  of  the  lateral  surface  of  the  pyra- 
mid. 

All  the  points  of  intersection  of  the  lateral  edges  with  the  cutting  plane 
are  seen  in  the  front  view,  and  may  be  projected  to  the  other  views.  The 
width  of  the  side  view  is  the  distance  JT-JT.  The  lateral  edges  7-7  and 
^t-7  appear  their  real  length  in  the  front  view  ;  therefore  the  distance  from 
the  vertex,  7,  to  the  points  c  and  d  of  the  section,  may  be  measured  from 
?F  to  CF  and  from  ?F  to  dF. 

To  obtain  the  distances  from  the  vertex  7  to  the  points  of  the  section  in 
the  other  lateral  edges,  draw,  in  the  front  view,  horizontal  lines  through 
points  a  and  b  to  the  left  or  the  right  lateral  edge,  and  measure  the  distances 
from  point  7  upon  these  lines,  as  explained  under  Fig.  89. 

FIG.  96.  Top  and  front  views  of  a  cone,  showing  the  section  made  by 
a  plane  passing  through  its  vertex. 

The  line  IF-3F  is  the  real  length  of  the  line  represented  by  iT-aT.  Line 
iF~3F  is  equal  to  the  altitude,  and  line  2T-3T  equal  to  the  base  of  the 
isosceles  triangle  which  is  the  section  made  by  the  plane.  Art.  161. 

FIG.  97.  Top  and  front  views  of  a  vertical  cone  intersected  by  a  plane 
parallel  to  the  front  plane  and  in  front  of  the  axis  of  the  cone;  also  the 
development  of  the  lateral  surface. 

The  circle  of  the  base  is  cut  in  points  7  and  2,  which  are  determined  in 
the  top  view.  To  obtain  j,  the  highest  point  in  the  section,  measure 
VT-3T,  the  distance  of  the  plane  from  the  axis  of  the  cone,  and  draw  a 
vertical  line  3' -3"  the  distance  VT-3T  from  the  axis.  This  represents  a 
side  view  of  the  plane,  and  intersecting  the  element  of  the  cone  at  3",  gives 
the  level  of  the  highest  point. 

To  obtain  two  other  points,  intersect  the  cone  by  a  horizontal  plane  B. 
This  gives  a  circle  which  appears  a  circle  in  the  top  view,  where  it  inter- 
sects the  cutting  plane  in  4T  and  j7',  points  in  the  required  line  of  intersec- 
tion ;  <f.F  and  5F  must  be  in  BF  and  in  projecting  lines  from  these  points 
of  the  top  view. 

Points  4.  and  5  are  in  circle  B,  which  is  the  base  of  a  cone,  just  as  / 
and  2  are  in  the  base  of  the  original  cone.  Any  number  of  planes  parallel 
to  B  may  be  taken,  and  each  will  give  two  points  in  the  section,  which  is  an 
hyperbola. 


136  MECHANICAL   DRAWING. 

To  develop  the  lateral  surface  of  the  cone  proceed  as  explained  in  Art. 
58  and  under  Fig.  88.  To  show  the  line  of  intersection  upon  it,  measure, 
upon  the  arc,  the  distance  of  IT  and  2T  from  bT,  and  set  this  distance  off 
from  bD  in  the  development.  The  circle  given  by  plane  B  becomes,  in  the 
development,  an  arc  parallel  with  the  one  which  bounds  the  lateral  surface 
of  the  cone.  In  the  circle  of  plane  B  are  points  4.  andjy  their  distances 
from  each  other  and  from  C  are  seen  in  the  top  view.  These  points  will  be 
placed  in  the  development  by  measuring  the  distance  C~4T  and  C-JT  on  the 
arc  of  the  top  view,  and  setting  it  off  from  C  in  the  development.  In  this 
way  all  the  points  by  which  the  section  is  obtained  may  be  placed  in  the 
development. 


PLATE  XI. 


Fig.  97. 


Fig.  96. 


MECHANICAL   DRAWING 


PLATE   XII. 

FIG.  98.  Top,  front,  and  left  side  views  of  a  -vertical  cone  intersected 
by  a  plane  at  an  angle  "with  its  base  and  cutting  all  the  elements;  the  real 
shape  of  the  section j  also  the  development  of  the  lateral  surface  of  the 
cone. 

The  points  in  which  the  contour  elements,  V-i  and  V-?,  of  the  front  view 
intersect  the  cutting  plane  are  seen  in  the  front  view  and  may  be  projected 
to  the  top  and  side  views.  The  points  f  and  g,  in  which  the  contour  ele- 
ments, V-j.  and  V-io,  of  the  side  view  intersect  the  cutting  plane  are  seen  in 
the  front  view,  and  from  this  view  may  be  projected  to  the  side  view.  To 
obtain  f  and  g  in  the  top  view  the  distance  between  these  points  may  be 
measured  in  the  side  view  and  then  set  off  in  the  top  view  ;  or  a  horizontal 
cutting  plane  may  be  taken  through  f  and  g;  this  gives  a  circle  which  in 
the  top  view  gives/"2"  zndgT.  To  obtain  other  points  in  the  section,  other 
horizontal  cutting  planes  may  be  used,  as  explained  under  Fig.  97,  or  equi- 
distant elements  may  be  placed  on  the  cone,  and  their  intersections  seen  in 
the  front  view.  As  the  surface  is  to  be  developed,  the  latter  method  is 
preferable.  The  lengths  of  these  elements  are  found  as  if  they  were  the 
edges  of  the  pyramid  of  Fig.  95.  The  lateral  surface  should  be  divided  into 
at  least  twelve  equal  parts. 

To  obtain  in  the  development  the  direction  of  the  curve  at  its  termina- 
tion on  lines  VD-ID,  add  one  twelfth  at  either  end  of  the  arc  which  is  the 
development  of  the  base  of  the  cone,  and  obtain  lines  VD—i2' and  VD—2*. 
From  VD,  on  these  lines  set  off  the  distance  VDbD  and  VDcD.  Trace 
the  curve  of  intersection  through  these  points. 

FIG.  99.  Top  and  front  views  of  a  cone  cut  by  a  plane  parallel  to  an 
element;  also  the  real  shape  of  the  parabola  which  is  the  section. 

This  section  may  be  obtained  as  explained  under  Figs.  97  or  98.  The 
drawing  makes  use  of  the  cutting  plane  explained  under  Fig.  97. 

FIG.  100.  Top,  front,  and  right  side  views  of  a  sphere,  intersected  by 
a  vertical  cone,  and  a  horizontal  cylinder. 

In  this  figure  the  axes  of  the  cylinder  and  cone  pass  through  the  centre 
of  the  sphere  ;  therefore  the  lines  of  intersection  are  circles.  The  planes 
of  these  circles  are  perpendicular  to  the  front  plane  and  are  therefore 
represented  by  straight  lines  in  the  front  view.  The  circles  in  which  the 
cone  and  sphere  intersect  are  parallel  to  the  top  plane  and  are  represented 
by  circles  in  the  top  view. 


PLATE  XII. 


12. 


II        12          Is          23 


2       3 


5        6 


Fig.  99. 


Fig.  100. 


140  MECHANICAL   DRAWING. 


PLATE    XIII. 

FIG.  101.  Top  and  front  views  of  an  elbow  extending  from  a  conical 
support, 

If  a  cylinder  or  cone  is  cut  by  a  plane  oblique  to  its  axis,  one  part  may  be 
revolved  upon  the  other  until  the  ellipses  of  the  parts  coincide  in  a  second 
position.  This  will  happen  when  an  arc  of  180°  has  been  formed  by  the 
revolution  of  the  parts.  After  such  revolution  the  angle  between  the  axes 
of  the  t\\o  parts  is  twice  that  of  the  angle  of  the  cutting  plane  and  the  axis 
of  the  original  solid. 

The  elbow  illustrated  may  be  formed  by  cutting  a  cylinder  at  45°  to  its 
axis,  and  revolving  the  parts  as  explained. 

The  plane  of  the  section  is  perpendicular  to  the  front  plane  and  there- 
fore the  ellipse  in  which  the  vertical  and  horizontal  cylinders  intersect  ap- 
pears a  straight  line  in  the  front  view  ;  it  appears  a  circle  in  the  top  view, 
for  the  surface  of  the  vertical  cylinder  is  perpendicular  to  the  top  plane. 

To  develop  the  cylinder,  divide  its  surface  into  any  number  of  equal 
parts  by  elements.  To  do  this  in  the  case  of  the  horizontal  cylinder  the 
circles  must  be  revolved  to  appear  their  real  shapes  and  divided  as  shown 
by  the  dotted  lines.  The  elements  having  been  drawn,  their  real  lengths 
are  seen  in  both  views  and  may  be  set  off  upon  the  respective  lines  of  the 
development. 

When  a  cylinder  is  to  be  developed  neither  of  whose  bases  is  at  right 
angles  to  the  axis  of  the  cylinder,  it  is  necessary  to  assume  a  circle  at  right 
angles  to  the  axis ;  this  will  develop  into  a  straight  line.  Thus,  suppose  the 
horizontal  part  to  be  represented  in  the  front  view  by  (a  b  de  a)F.  To 
develop  this  cylinder,  it  will  be  necessary  to  develop  a  circle  which  is  at 
right  angles  to  the  axis.  This  circle  may  be  assumed  anywhere,  as  at  a  c, 
and  develops  into  a  straight  line  aDcD.  The  development  of  the  part  at 
the  right  of  the  circle  should  be  placed  at  the  right  of  aDcD,  just  as  the 
part  at  the  left  is  placed  at  the  left  of  aDcD. 

The  frustum  of  the  cone  is  developed  as  explained  under  Fig.  88. 

FIG.  1 02.  Top  and  front  views  of  an  upright  square  prism  inter- 
sected by  a  horizontal  square  prism  at  the  right,  and  by  a  triangular 
prism  at  the  left;  also  the  developments  of  the  lateral  surfaces  of  the 
vertical  and  triangular  prisms,  and  half  that  of  the  horizontal  sqttare 
prism. 

The  points  in  which  all  the  edges  of  both  horizontal  solids  intersect  the 
surface  of  the  vertical  one  are  seen  in  the  top  view  and  are  readily  pro- 
jected to  the  front  view. 


142  MECHANICAL   DRAWING. 

Develop  the  lateral  surface  of  the  upright  prism  as  explained  in  Arts. 
51  and  54.  The  horizontal  faces  of  the  square  block  intersect  the  upright 
prism  in  horizontal  lines  which  are  parallel  to  its  bases,  and  which  in  the 
development  must  be  parallel  to  MN  and  OP.  The  vertical  faces  inter- 
sect the  prism  in  vertical  lines  parallel  to  the  edges  B  and  Dj  and  the  line 
of  intersection  of  the  horizontal  and  vertical  square  prisms  on  the  develop- 
ment of  the  vertical  prism  is  a  rectangle  whose  length  is  the  distance 
iT-fT-^Tj  its  width  is  the  distance  if-2.f. 

The  horizontal  face  of  the  triangular  prism  intersects  the  vertical  square 
prism  in  lines  which  develop  into  a  straight  line  parallel  to  OP;  its 
length  is  the  distance  7-5-6  of  the  top  view ;  the  positions  of  this  line  and 
also  that  of  point  5  are  seen  in  the  front  view. 

The  other  developments  require  no  special  explanation.  In  all  the  prob- 
lems it  is  simply  necessary  to  remember  that  the  development  gives  the  real 
length  of  every  line  of  every  surface  and  that  the  length  of  any  line  must  be 
taken  from  the  view  in  which  it  appears  its  real  length,  or,  if  not  seen  of  its 
real  length  in  any  view,  this  must  be  determined  as  explained  in  Art.  114. 

The  development  of  any  surface  is  most  easily  obtained  by  placing  it 
so  that  one  set  of  dimensions  may  be  projected  from  the  front  view. 


PLATE  XIII. 


144 


MECHANICAL   DRAWING. 


PLATE    XIV. 


FIG.  103.  An  upright  square  prism  intersected  at  the  left  by  a  hori- 
zontal square  prism,  whose  lateral  faces  are  at  ^j°  to  the  top  and  front 
planes,  and  at  the  right  by  an  hexagonal  prism  -with  two  of  its  lateral 
faces  vertical;  also  the  development  of  the  lateral  surface  of  the  hexagonal 
prism,  and  half  that  of  the  horizontal  square  prism. 

The  intersections  are  found  as  previously  explained. 

To  develop  the  lateral  surfaces  of  the  penetrating  prisms,  measure  the 
widths  of  the  faces  in  the  end  views,  and  the  lengths  of  the  lateral  edges 
in  the  front  or  top  view,  and  combine  these  dimensions  as  explained. 

FIG.  104.  An  upright  square  prism  intersected  at  the  right  by  a  hori- 
zontal hexagonal  prism,  two  lateral  faces  being  horizontal,  and  at  the  left 
by  a  horizontal  cylinder;  also  the  developments  of  the  lateral  surface  of  the 
square  prism,  and  half  those  of  the  hexagonal  prism  and  of  the  cylinder. 

In  the  front  view,  the  intersections  of  the  upper  and  lower  elements 
/  and  7  of  the  cylinder  with  the  left  edge  of  the  prism  are  seen.  In  the  top 
view,  the  intersections  of  the  front  and  back  elements  4.  and  10  are  seen. 
To  obtain  other  points  assume  elements,  to  place  which  a  side  view  is  nec- 
essary ;  when  determined  in  this  view  they  may  be  transferred  to  the  top 
view  by  revolving  the  circle  in  the  top  view  until  it  appears  a  circle, 
dividing  it,  and  numbering  the  points,  to  correspond  with  the  end  view ;  or 
projection  methods  may  be  used  and  the  distances  of  the  points  2,  12,  etc., 
from  a  vertical  line  through  the  centre  of  the  cylinder  may  be  taken  in  the 
compasses  from  the  side  view  and  set  off  in  the  top  view,  one  half  each  side 
of  the  axis. 

The  intersections  of  the  elements  and  the  prism  are  seen  in  the  top  view 
and  may  be  projected  to  the  front  view. 

The  intersections  of  the  Cylinder  and  the  faces  of  the  prism  are  semi- 
ellipses  which  appear  a  semi-circle  in  the  front  view. 

The  intersection  of  the  hexagonal  and  square  prisms  is  found  as  explained. 

To  develop  the  lateral  surface  of  the  square  prism  and  show  the  lines 
of  intersection  upon  it,  first  develop  the  entire  lateral  surface  ;  then  draw,  in 
the  front  view,  verticals  upon  the  surface  of  the  square  prism  through  the 
points  of  the  intersections.  Place  these  lines  in  the  development  by  setting 
off  from  AD  and  CD  the  distances,  seen  in  the  top  view,  of  the  lines  from 
AT  and  CT;  thus  to  obtain  aD  and  eD  draw  a  parallel  to  AD  at  the  distance 
ATaT  from  it,  and  then  set  off  in  this  line  the  distances  of  a?  and  e**  from 
AFCF.  In  this  way  all  the  points  necessary  to  determine  the  lines  of  inter- 
section in  the  development  may  be  obtained  and  the  lines  drawn  through 
the  points. 


PLATE  XIV. 


r 


146  MECHANICAL   DRAWING. 


PLATE    XV. 

FIG.  105.  Top  and  front  views  of  an  upright  cylinder  intersected  at 
the  left  by  a  horizontal  square  prism,  and  at  the  right  by  a  horizontal 
cy  Under  j  also  the  development  of  the  lateral  surface  of  the  cylinder  and 
half  that  of  each  intersecting  body. 

Both  intersections  and  developments  are  the  same  in  principle  as  those 
already  explained.  The  line  in  which  the  square  prism  intersects  the  cylin- 
der develops  into  a  rectangle  in  the  development  of  the  cylinder,  because 
the  horizontal  faces  intersect  the  cylinder  in  circles  which  develop  into 
straight  lines,  and  the  vertical  faces  intersect  elements  of  the  cylinder. 

FIG.  1 06.  Top  and  front  views  of  a  vertical  cylinder  intersected  at  the 
left  by  a  square  prism,  and  at  the  right  by  a  triangular  prism  at  45°  with 
the  cylinder;  also  developments. 

In  the  front  view  the  distance  EFFF  is  equal  to  the  distance  E'k  of 
the  view,  F'E'G',  of  the  end  of  the  prism,  which  must  be  drawn  before  the 
front  view  can  be  completed.  It  is  not  necessary  to  make  this  separate 
drawing,  as  the  dotted  lines  in  the  front  view  which  show  half  the  end  give 
all  that  is  necessary. 

The  points  in  which  the  lateral  edges  of  the  prisms  intersect  the  cylin- 
der are  seen  in  the  top  view.  The  inclined  sides  of  the  prisms  intersect  the 
cylinder  in  ellipses,  points  in  which  may  be  found  by  assuming  lines  on  the 
surfaces  of  the  penetrating  solids.  A  line  from  IT  intersects  the  cylinder 
at  aT,  from  which  the  positions  of  aF  and  CF  may  be  obtained.  A  line  from 
dT  gives  the  position  of  gT.  In  this  way  any  number  of  points  may  be  ob- 
tained. 

The  development  of  all  the  lateral  surface  of  the  triangular  prism  is 
given  ;  point  g  is  in  a  parallel  to  the  lateral  edges,  and  distant  from  E  the 
distance  EFd'j  point  n  is  in  a  parallel  to  the  lateral  edges,  whose  distance 
from  F  is  the  distance  FTeT. 

In  the  development  of  the  square  prism,  points  aD  and  CD  are  in  par- 
allels to  the  lateral  edges,  and  the  distances  Bs-is  and  BS-2S  from  BD. 

In  the  development  of  the  line  of  intersection  upon  the  cylinder,  point 
gD  is  in  a  parallel  to  fE>mD,  whose  distance  from  it  is  seen  in  the  top  view, 
fromy?*  to  gT,  measured  on  the  arc.  Point  nD  is  in  this  same  line,  and 
hD  is  in  a  parallel  \.ofDmD,  whose  distance  from  it  is  the  distance  fTgThT, 
measured  on  the  arc. 


PLATE  XV. 


Fig.  105 


Fig.  106. 


148  MECHANICAL   DRAWING. 


PLATE    XVI. 

FIG.  107.  Front,  top,  and  right  side  views  of  a  horizontal  square 
prism  intersected  by  a  -vertical  square  pyramid;  also  developments. 

The  intersections  of  lines  A  V  and  CV  with  the  edges  of  the  prism  are 
seen  in  the  front  view,  and  the  intersections  of  lines  B  V  and  DV  with  the 
prism  are  seen  in  the  side  view.  Project  these  points  of  intersection  from 
the  side  to  the  front  view  and  from  the  front  to  the  top  view.  The  inter- 
secting surfaces  are  plane  and  therefore  intersect  in  straight  lines,  which 
join  the  points  of  intersection. 

To  obtain  the  lines  of  intersection  in  the  development  of  the  pyramid, 
measure  the  distances  from  the  vertex  V  to  the  points  of  intersection  in  VA 
and  VC  in  the  front  view,  and  the  distances  from  the  vertex  to  the  points 
of  intersection  in  VB  and  VD  in  the  side  view.  Set  these  distances  off  on 
the  proper  lines  of  the  development  and  join  the  points. 

To  obtain  the  points  for  the  lines  of  intersection  upon  the  lateral  sur- 
face of  the  prism  place  points  k  and  d  in  edge  2,  and  e  and  f  in  edge  4; 
their  positions,  in  the  edges,  are  seen  in  the  front  view.  Points  a,  h,  g,  and  b 
are  in  a  plane  perpendicular  to  the  axis  of  the  prism  and  must,  in  the  devel- 
opment, be  in  a  straight  line  perpendicular  to  the  lateral  edges.  The  dis- 
tances of  a  and  h  from  edge  2,  and  of  b  and  g  from  edge  4  are  seen  in  the 
side  view. 

FIG.  1 08.  Top,  front,  and  left  side  views  of  a  horizontal  cylinder  in- 
tersected by  an  upright  square  pyramid ';  also  developments. 

This  problem  is  solved  in  the  same  way  as  Fig.  107,  but  the  intersec- 
tions are  not  straight  lines.  Each  lateral  face  of  the  pyramid  cuts  the  cyl- 
inder in  lines  which  are  parts  of  the  complete  ellipse  which  will  be  given  by 
the  plane  of  each  face  cutting  through  the  cylinder. 

The  intersections  of  the  lateral  edges  are  seen  in  the  front  and  side 
views.  To  obtain  other  points  in  the  lines  of  intersection  assume  any  line, 
as  V-i,  on  the  surface  of  the  pyramid.  This  line  intersects  the  surface  of 
the  cylinder  in  points  b  and  g;  these  points  are  readily  obtained  in  the  front 
and  top  views  from  bs  and  gs.  In  the  side  view  V-i  also  represents  the  line 
V-2  on  the  right  face  of  the  pyramid  in  which  points  h  and  o  are  situated. 
In  this  way  any  number  of  points  in  the  curves  may  be  found. 

To  develop  the  pyramid  and  show  the  line  of  intersection,  the  distance 
from  V 'to  the  points  of  the  intersection  in  VD  and  VB  must  be  measured 
in  the  front  view  ;  the  distances  from  V  to  the  points  of  the  intersection  in 
VA  and  VC  must  be  measured  in  the  side  view. 


MECHANICAL   DRAWING. 

To  place  the  points  ft,  g  and  0,  h  in  lines  V-i  and  V-2,  draw  through  these 
points  lines  parallel  to  the  base  of  the  pyramid  ;  the  line  through  o  inter- 
sects VFBF  in  point  4F.  The  distance  of  4  from  V  is  seen  in  the  front 
view  and  4°  is  readily  placed.  A  line  through  <f?>  parallel  to  ADBD  and 
intersecting  VD-2D  must  give  OD;  in  this  way  all  the  points  may  be  found. 

To  obtain  the  lines  of  intersection  in  the  development  of  the  cylinder, 
measure,  in  the  end  view  and  around  the  arc,  the  distances  apart  of  the  ele- 
ments containing  the  points  ;  determine  the  positions  of  the  points  in  the 
elements  in  the  front  view. 


PLATE  XV?. 


Fig.  108. 


152  MECHANICAL   DRAWING. 


PLATE   XVII. 

FIG.  109.  Top  and  front  views  of  a  vertical  cone  intersected  at  the 
left  by  a  horizontal  triangular  prism,  and  at  the  right  by  a  sguare  prism. 

The  points  in  which  edge  D  of  the  triangular  prism  and  edges  2  and  4 
of  the  square  prism  intersect  the  cone  are  seen  in  the  front  view,  for  these 
edges  are  in  the  plane  of  the  contour  elements.  To  find  other  points  in  the 
curves  in  which  the  cone  is  intersected  by  the  lateral  faces  of  the  prisms, 
assume  any  horizontal  cutting  plane,  as  AB ;  this  gives,  upon  the  triangular 
prism,  a  rectangle  whose  width  is  7-8,  and  upon  the  square  prism,  one  whose 
width  is  J-6j  the  plane  intersects  the  cone  in  a  circle  C.  The  circle  and  these 
rectangles  intersect  in  points  e,  d  and  a,  b,  which  must  be  points  in  the  lines 
of  intersection  of  the  cone  with  the  triangular  and  square  prisms.  As  many 
points  as  are  desired  may  be  obtained  by  means  of  other  cutting  planes. 

FIG.  no.  Top  and  front  views  of  a  vertical  cone  intersected  by  a 
horizontal  sguare  pyramid. 

Edges  E  and  F  intersect  the  contour  elements  of  the  cone.  To  find  the 
intersections  of  edges  G  and  H,  and  also  to  obtain  other  points  in  the  inter- 
sections, intersect  the  solids  by  cutting  planes.  CD  is  a  horizontal  cutting 
plane  and  gives  four  points,  e,f,g,  h,  in  the  lines  of  intersection. 

FIG.  in.  A  vertical  cylinder  intersected  by  a  horizontal  cone;  also 
developments. 

The  principles  involved  have  been  previously  explained. 

To  develop  the  cone,  find  the  distances  of  points  b,  c,  d,  e,  andy  from 
the  vertex  V,  by  measuring  upon  the  contour  elements,  as  explained  under 
Fig.  98. 

Obtain  the  line  of  intersection  in  the  development  of  the  cylinder  by 
measuring,  in  the  top  view  and  around  the  arc,  the  distances  between  the 
elements  containing  the  points,  and  by  measuring,  in  the  front  view,  the 
positions  of  the  points  in  these  elements. 


PLATE  XVII. 


SECTION  AT  A  B 


SECTION  AT  C  D 


154  MECHANICAL   DRAWING. 


PLATE   XVIII. 

FIG.  1 12.  Front,  top,  and  left  side  "views  of  a  loose  joint  hinge ;  also 
front  and  top  -views  of  the  two  parts  forming  the  hinge,  when  they  are 
separated  from  each  other. 

FIG.  1 13.  Front,  top,  and  left  side  views  of  a  sash  lift;  also  a  section 
on  AS. 

FIG.  1 14.  Front,  bottom,  and  left  side  views  of  a  drawer  pull;  also  a 
section  on  CD. 

Most  of  the  objects  shown  on  Plates  XVIII  to  XXII  inclusive  do  not 
require  all  the  views  given  to  make  them  complete  working  drawings.  In 
Fig.  113  or  114,  for  instance,  simply  the  front  and  top  views  and  the  sec- 
tion furnish  all  the  information  necessary  to  make  the  object.  The  side 
views  are  given  to  familiarize  the  student  with  the  arrangement  of  all  the 
views. 


PLATE  XVIII. 


Fig.  112. 


LOOSE  JOINT  HINGE. 


Fig.  113. 


SASH  LIFT 


SECTION  AT  A  B 


fig.  114. 


DRAWER  PULL. 
Pi 


SECTION  AT  C  D 


156  MECHANICAL   DRAWING. 


PLATE    XIX. 

FIG.  115.  Front,  top,  and  right  side  views  of  an  iron-cased  bolt,  with 
details  of  the  same. 

Dotted  lines  are  very  confusing  when  numerous  ;  therefore  when  many 
would  be  required  it  is  customary  to  represent  the  different  parts  of  an  ob- 
ject separately. 

In  this  figure  all  the  views  are  needed  to  make  the  construction  clear. 

The  ornamental  tracery  upon  the  object  is  not  represented.  Such  de- 
tails should  be  omitted  when  pupils  make  working  drawings  of  common 
objects. 


PLATE  XIX. 


Fig.  115. 


DETAILS  OF  AN  IRON  CASED  BOLT. 

THREE   FOURTHS   SIZE. 
DRAWING   19      BOSTON   DEC.'94. 


ID 


SECTION  OF  CASE  ON  A  B 


1C 


SECTION  OF  CASE  ON  C  D 


SECTION  AT  E  F 


BRASS  SPRING 


SECTIONS  OF  STRIKER 


158  MECHANICAL   DRAWING, 


PLATE   XX. 

FIG.  1 1 6.     Details  of  an  iron  side  pulley. 

A  is  a  front  view  of  the  pulley. 

B  is  a  side  view  of  the  pulley. 

C  is  a  section  of  the  frame  on  GH. 

D  is  a  back  view  of  the  frame. 

E  is  a  front  view  and  a  section  on  EF  of  the  wheel. 

FIG.  117.     Front,  top,  and  side  views  of  an  iron  bracket. 

FIG.  1 1 8.     Front,  top,  and  bottom  views  of  an  iron  clamp. 


PLATE  XX. 


IRON  SIDE  PULLEY. 


Fig.  116. 


AT  E  F 


-*• 


SHELF  BRACKET. 


Fig.  117 


IRON  CLAMP. 


l6o  MECHANICAL   DRAWING. 


PLATE    XXI. 

FIG.  119.  Free-hand  sketches  of  an  iron  caster,  from  which  finished 
drawings  are  to  be  made. 

In  making  such  sketches  the  proportions  of  all  the  parts  should  be  rep- 
resented as  truly  as  possible ;  but  the  parts  should  be  drawn  free-hand, 
and  the  drawings  completed  before  the  object  is  measured.  It  is  not 
necessary  that  the  proportions  be  exact,  but  it  is  better  to  have  them  so,  as 
errors  in  measuring  or  in  figuring  the  sketches  may  be  made.  If  the 
sketches  are  correct  in  their  proportions,  these  errors  will  be  shown  when 
the  drawings  are  made  to  scale,  and  compared  with  the  free-hand  sketches. 


PLATE  XXI. 


SKETCHES 

OF   AN 

RON    CASTER. 


PLATE. 


FffAME. 


1 62  MECHANICAL  DRAWING. 


PLATE   XXII. 

FIG.  120.     Views  of  a  wooden  faucet. 

A  is  the  front  view. 

B  is  the  top  -view. 

C  is  the  left  side  view. 

D  is  a  section  at  KL. 

E  is  a  section  at  MN. 

The  plug  is  surrounded  by  cork,  which  gives  a  tight  joint.  The  cork  is 
shown  in  the  sections  by  the  dotted  parts. 

The  cut  surface  of  the  wood  may  be  shown  as  illustrated  on  page  61. 

The  lines  in  which  the  conical  parts  of  the  faucet  intersect  the  faces  of 
the  octagonal  part  are  the  only  points  requiring  explanation  ;  these  lines 
are  really  hyperbolas  (Fig.  97).  It  is  customary  to  represent  such  curves 
by  arcs  of  circles  of  which  the  highest  points  and  the  points  in  the  edges 
are  found  exactly  ;  the  lines  of  intersection  upon  the  narrow  faces  are 
so  slightly  curved  that  straight  lines  may  be,  and  generally  are,  substi- 
tuted for  the  arcs. 

To  determine  the  points  for  the  arcs,  notice  that,  in  the  front  view,  the 
contour  elements  of  the  conical  parts  must  intersect  the  upper  and  lower 
lines  at  the  central  or  highest  points  of  the  curves  in  two  wide  faces.  As  the 
wide  faces  are  of  equal  width,  the  highest  points  in  the  curves  of  these  faces, 
in  both  front  and  top  views,  will  be  in  vertical  lines  drawn  through  the 
points  of  intersection.  To  find  the  points  in  the  edges,  an  edge  must  be 
represented  so  that  its  real  distance  from  the  centre  of  the  faucet  shall  be 
seen  ;  to  do  this  revolve,  about  the  centre  of  the  end  view,  point  as  into  the 
position  a';  project  from  a'  a  horizontal  line  which  represents  an  edge,  and 
intersecting  element  2-1  of  the  cone  at  a",  gives  the  position  of  the  point 
where  the  conical  surface  cuts  the  edges  of  the  faucet.  These  edges  are 
equally  distant  from  the  centre  of  the  faucet ;  therefore  in  both  front  and 
top  views  all  the  arcs  must  end  at  the  edges  in  points  which  are  in  a  vertical 
line  drawn  through  a". 


o 


164  MECHANICAL   DRAWING. 


PLATE    XXIII. 

FIG.  121.     A  front  view  and  section  of  a  washer. 

FIG.  122.  Front  and  top  views  of  the  end  of  a  rod  adapted  to  hold  a 
lever,  which  is  pivoted  on  screws  inserted  one  in  each  part  of  the  fork. 

The  forked  part  is  cast  and  is  screwed  to  the  end  of  a  round  rod,  of 
which  a  part  is  shown. 

The  information  which  an  end  view  would  give  is  necessary  to  make 
the  drawings  give  all  the  facts  of  form,  but  the  placing  of  the  letter  D  after 
the  dimension  which  shows  the  diameter  of  the  round  part  of  the  fork, 
makes  the  end  view  unnecessary. 

Threaded  holes  are  best  shown  as  illustrated  ;  in  one  view  two  circles 
are  drawn,  the  outer  being  dotted  and  the  inner  full.  Some  draughtsmen 
represent  part  of  the  outer  circle  by  a  full  line  which  becomes  tangent  to 
the  inner  circle.  This  is  more  as  the  thread  appears,  but  the  simpler  rep- 
resentation is  the  better.  In  the  other  view  the  dotted  lines  which  show 
the  section  of  the  thread  are  at  an  angle  of  60°  with  each  other ;  they 
should  not  join  each  other,  as  this  will  destroy  the  effect  of  a  dotted  line  ; 
they  should  be  drawn  free-hand  with  a  writing-pen.  Only  the  outline  of  the 
thread  should  be  represented,  for  dotted  lines  in  place  of  all  the  full  lines 
shown  on  Plate  XXVI  will  give  an  unsatisfactory  drawing. 

FIG.  123.  A  top  view  and  a  front  view,  one  half  of  which  shows  a 
section  through  the  centre,  of  a  machine  detail  called  a  gland,  and  used  for 
compressing  packing  about  a  piston  to  prevent  the  escape  of  steam  or 
water. 

It  is  customary  to  show  in  section  one  half  (or  any  other  part)  of  the 
view  of  a  symmetrical  object  ;  this  saves  the  time  required  to  make  com- 
plete views  of  the  outside  and  of  the  inside  of  the  object. 

In  any  working  drawing  no  more  lines  should  be  given  than  are  neces- 
sary to  show  the  construction  ;  dotted  lines  are  especially  to  be  avoided. 


1 66  MECHANICAL  DRAWING. 


PLATE    XXIV. 

FIG.  124.  Top  and  side  views,  and  a  front  view,  one  half  of  which 
shows  a  section,  of  a  bearing  for  the  end  of  a  shaft  which  moves  back  and 
forth  in  a  plane,  and  so  requires  a  bearing  that  rotates.  The  shaft  rests 
upon  a  box  of  composition  B,  which  is  replaced  when  worn. 

FIG.  125.  A  front  view,  of  which  one  half  is  in  section,  and  a  verti- 
cal cross  section,  of  a  bearing  used  upon  a  locomotive. 


PLATE  XXIV. 


SHAFT  BEARINGS. 

SCALE,  3  in.  =  I  ft. 
DRAWING  24.     BOSTON.  DEC    94. 


„- 


: W j 

_3*'_  -3i- J 


1 68  MECHANICAL   DRAWING. 


PLATE    XXV. 

THE    HELIX. 

If  a  point  moves  in  two  directions  about  a  given  line  as  axis,  —  that  is 
around  and  along  the  line  at  the  same  time,  —  a  helical  curve  results. 

The  simplest  form  of  this  curve  is  found  in  a  common  spring.  The 
lines  which  bound  the  threads  of  screws  are  helices  ;  these  curves  are  found 
in  many  constructions. 

The  motion  of  the  point  in  its  different  directions  may  be  uniform  or 
variable ;  in  the  common  forms  of  the  curve  both  motions  are  uniform,  and 
are  produced  by  a  point  which  moves  uniformly  around  and  along  a  cylin- 
der at  the  same  time. 

The  distance  which  the  point  travels  along  the  cylinder,  in  going  once 
around  it,  is  called  the  pitch. 

To  draw  the  common  helix,  divide  the  circle  which  is  a  view  of  the  line 
in  which  the  generating  point  moves  about  the  axis,  into  any  number  of 
equal  parts,  and  divide  the  pitch  into  the  same  number  of  equal  parts. 
When  the  point  has  moved  upon  the  circle  of  the  end  view  over  one  of 
the  equal  spaces  into  which  the  circle  is  divided,  it  has  moved  in  the  other 
view,  along  the  axis  from  its  starting  point,  a  distance  equal  to  one  of  the 
equal  spaces  into  which  the  pitch  is  divided  ;  when  it  has  moved  one 
quarter  around  the  circle  it  has  moved  one  quarter  of  the  pitch  ;  when  it 
has  moved  one  half  around  the  circle  it  has  moved  one  half  of  the  pitch, 
and  so  on  for  the  whole  revolution  of  the  generating  point.  Hence  to  ob- 
tain all  the  points  for  the  view  of  the  helix,  draw  parallels  to  the  axis  from 
the  points  of  division  in  the  circle,  and  intersect  these  lines  by  perpendicu- 
lars to  the  axis  from  the  equal  divisions  of  the  pitch.  If  the  points  of  the 
circle  and  the  pitch  are  numbered  in  order  from  the  first  point,  the  inter- 
sections of  lines  of  the  same  number  will  be  points  in  the  required 
curve. 

FIG.  1 26.     A  helical  curve  upon  a  cylinder. 

The  curve  is  found  as  explained  above.  The  division  of  the  pitch  into 
twelve  equal  parts  is  advisable  for  pupils.  To  avoid  error,  the  points  of  the 
bottom  view  and  also  those  of  the  pitch  may  be  numbered  from  i  to  12. 
The  curve  is  symmetrical;  its  vertices  12  and /"must  be  curved. 

If  the  cylinder  is  developed,  the  helix  will  become  the  hypothenuse  of  a 
right-angled  triangle  A,  of  which  one  side  of  the  right  angle  is  equal  to  the 
circumference  of  the  cylinder  and  the  other  is  equal  to  the  pitch. 

When  many  curves  of  the  same  pitch  are  desired,  a  templet  should  be 


170  MECHANICAL   DRAWING. 

made  of  thin  wood  to  fit  the  first  one  drawn  ;  by  this  the  others  can  be 
quickly  and  accurately  drawn. 

FlG.  127.  A  cylinder  with  a  helical  blade  or  surface,  formed  by  the 
revolution  of  a  line  perpendicular  to  the  cylinder  around  and  along  the 
cylinder  at  the  same  time. 

The  two  ends  of  the  generating  line  describe  helices  of  the  same  pitch, 
but  on  cylinders  of  different  diameters.  Each  helix  is  found  as  already  ex- 
plained. 

Instead  of  revolving  from  left  to  right  the  line  revolves  from  right  to 
left,  and  the  helices  advance  along  the  cylinder  in  the  opposite  direction  to 
that  of  the  helix  in  Fig.  1  26. 

Threads  are  called  right-handed  when  they  advance  as  in  Fig.  1  26,  and 
left-handed  when  they  advance  as  in  Fig.  127.  The  most  common  ex- 
amples of  both  kinds  of  threads  are  found  on  the  axles  of  all  wagons.  The 
threads  on  the  right  hand  ends  are  right-handed  ;  the  threads  on  the  left 
hand  ends  are  left-handed,  that  is,  the  nuts  are  screwed  upon  them  by  the 
motion  which  unscrews  those  on  the  right  hand  ends. 

A  helical  curve  may  be  generated  by  a  point  which  moves  along  the 
surface  of  a  cylinder  at  a  varying  rate  of  speed,  while  its  motion  around  the 
cylinder  is  uniform  ;  or  it  may  be  generated  by  a  point  whose  motion  in 
both  directions  is  variable.  A  helical  curve  may  be  traced  upon  a  conical 
or  spherical  surface.  A  helical  surface  may  be  generated  by  a  line  which 
moves  as  in  Fig.  127,  or  by  a  line  which  is  inclined  to  the  axis  of  the  cylinder. 
These  problems  are  of  too  advanced  a  nature  to  be  given  in  this  book. 

FIG.  128.  A  V-threaded  bolt  and  nut,  the  upper  part  of  the  bolt  and 
the  nut  showing  a  section  through  the  centre. 

The  section  of  this  thread  is  an  equilateral  triangle.  The  size  of  the 
thread  depends  upon  the  diameter  of  the  bolt  ;  there  are  regular  standard 
threads  for  bolts  of  all  diameters,  and  other  standard  threads  for  pipes. 
Iron  bolts  of  the  following  diameters  have  threads  as  specified  : 

Diameter  of  Bolt.  Threads  per  inch. 

%  inch      ......         20 

#    "         ......         16 

"  ......  12 


i    "         ......          8 

i#    "  6 

We  will  suppose  the  screw  shown  to  be  of  wood  ;  the  thread  is  there- 
fore large  in  proportion  to  the  bolt.  First  draw  the  section  of  the  lower 
thread  and  divide  the  pitch  into  twelve  equal  parts.  As  only  half  of  the 


172  MECHANICAL   DRAWING. 

bottom  view  is  shown,  this  is  divided  into  six  equal  parts.  The  outer  and 
inner  helices  are  obtained  as  explained  under  Fig.  126,  and  the  drawing  of 
the  bolt  is  completed  by  drawing  the  section  of  the  thread  which  connects 
the  outer  and  inner  helical  curves. 

In  exact  drawings  of  bolts  whose  threads  are  large,  the  F-shaped  outline 
of  the  section  of  the  thread  will  come  inside  the  correct  projection  of  the 
helical  surfaces  forming  the  thread ;  but  to  obtain  the  exact  projection  is 
so  complicated  a  problem  that  generally  the  thread  is  represented  as  illus- 
trated. A  straight  line  tangent  to  the  two  helices  approximates  the  actual 
projection.  As  the  helices  are  not  angular  at  their  vertices  it  is  sometimes 
necessary  to  represent  the  thread  in  this  way  instead  of  by  the  line  of  the 
section  which  is  given  in  the  figure. 

When  the  first  outer  and  inner  helices  are  complete,  the  others  may 
be  obtained  by  setting  off  the  pitch,  with  the  dividers,  from  the  points  of 
the  first  lines  as  many  times  as  other  curves  are  desired. 

The  lines  of  the  thread  in  the  interior  of  the  nut  have  the  opposite 
direction  to  those  upon  the  bolt ;  they  correspond  to  the  dotted  lines  of  the 
bolt  and  are  obtained  in  the  same  way. 


PLATE  XXV. 


Fig.  126. 


PITCH 


Fig.  127. 


174  MECHANICAL   DRAWING. 


PLATE   XXVI. 

FIG.  r  29.  A  square-threaded  bolt  and  nut,  the  upper  part  of  the  bolt 
and  the  nut  being  in  section. 

In  the  square  thread  both  the  thread  and  the  space  are  square,  each 
occupying  one  half  the  pitch.  There  are  two  helical  curves  at  the  inside  of 
the  thread,  and  two  at  the  outside  ;  the  points  for  these  curves  are  found  as 
already  explained. 

Figs.  1 28  and  1 29  give  the  actual  projections  of  the  common  forms  of 
threads.  These  threads  are  cut  in  a  lathe,  and  drawings  for  them  are  un- 
necessary, as  all  that  the  workman  requires  is  to  know  the  diameter  of  the 
bolt,  the  pitch  of  the  thread,  and  its  shape.  These  facts  might  be  written 
upon  the  drawing,  but  it  is  the  custom  to  represent  threads  conventionally 
by  means  of  straight  lines,  which  require  less  time  to  draw  than  do  the 
curves. 

FIG.  130.  A  bolt  with  V  and  square  threads,  and  a  nut  in  section  for 
each. 

At  the  right  a  F-threaded  bolt  and  nut  are  shown.  The  drawing  is 
given  by  substituting  straight  lines  for  the  curves  of  Fig.  1 28. 

Two  methods  of  representing  the  thread  of  a  square-threaded  bolt  and 
nut  are  shown  at  the  left.  The  threads  at  the  left  end  of  the  bolt  and  in 
the  nut  are  obtained  by  substituting  straight  lines  for  the  curves  of  Fig.  1 29. 
The  threads  near  the  centre  of  the  bolt  are  shown  more  simply  by  straight 
lines  representing  only  the  outer  edges  of  the  thread. 

FIG.  131.     Three  different  ways  of  representing  a  V  thread. 

At  A  is  shown  a  method  suitable  for  use  when  the  screw  is  very  small ; 
the  lines  of  the  bolt  in  such  a  case  may  be  omitted,  especially  when  the  end 
of  the  bolt  shows  for  a  little  distance  outside  of  its  nut. 

At  B  is  shown  the  representation  used  by  many  draughtsmen.  Some- 
times the  long  lines  are  made  heavy  instead  of  the  short  lines.  Heavy 
short  lines  give  the  best  effect  of  the  thread. 

At  C  is  shown  a  satisfactory  representation,  which  requires  less  time  to 
produce  than  the  other  methods. 

It  is  not  necessary  that  there  be  as  many  long  lines  per  inch  as  there 
are  threads,  though  it  is  well  for  pupils  to  represent  each  thread  by  a  long 
line,  because  they  are  not  able  to  space  by  eye  so  as  to  produce  a  satisfac- 
tory result. 

The  angle  at  which  the  lines  of  these  conventional  representations  are 
drawn,  should  be  determined  by  the  spacing  of  the  lines.  In  all  common 


iy6  MECHANICAL   DRAWING. 

threads  point  j  of  Fig.  131  should  be  half-way  between  points  /  and  2. 
This  drawing  represents  correctly  a  single-threaded  right-handed  bolt.  The 
opposite  direction  of  the  long  line,  that  is  from  j  to  .?,  will  represent  a  left- 
handed  thread. 

FIG.  132.     A  triple-threaded  bolt. 

A  bolt  may  have  any  number  of  parallel  threads.  In  this  case  there 
are  two  threads,  B  and  C,  between  the  turns  of  the  thread  A.  One  turn  of 
the  bolt  moves  it  the  distance  of  the  pitch  AA;  the  threads  A,  J3,  and  C 
give  three  times  the  strength  that  would  be  given  by  thread  A.  To  draw 
this  bolt,  divide  the  pitch  into  as  many  equal  parts  as  there  are  threads,  and 
place  one  thread  in  each  space,  as  in  a  single-threaded  bolt. 

FIG.  133  shows  the  conventional  shading  sometimes  added  to  a  draw- 
ing representing  cylindrical  objects.  Part  of  Fig.  132  shows  how  a  vertical 
cylinder  is  shaded. 


PLATE  XXVI. 


178  MECHANICAL   DRAWING. 


PLATE    XXVII. 

FIG.  134.     Three  views  of  a  square  bolt-head. 

FIG.  135.  Three  views  of  a  square  bolt-head,  whose  corners  have  been 
chamfered  (bevelled). 

The  surface  of  the  chamfer  is  really  that  of  a  cone,  which  is  shown  by 
the  dotted  lines.  This  cone  will  be  intersected  by  each  of  the  four  faces  of 
the  head  in  an  hyperbola.  In  practice  this  is  never  drawn,  but  is  repre- 
sented by  an  arc  of  a  circle,  whose  points  are  found  as  follows  :  The  con- 
tour elements  of  the  cone  intersect  the  vertical  lines  which  represent  the 
right  and  left  sides  of  the  head,  and  give  the  highest  points  of  the  curves 
of  intersection.  To  obtain  the  lowest  points,  which  must  be  in  the  edges 
since  they  are  farther  from  the  centre  than  any  other  lines  of  the  faces, 
revolve  2T  to  21  and  draw  from  2  a  vertical  line  to  intersect  the  element  of 
the  cone  at  2".  This  vertical  represents  an  edge  of  the  head,  when  its  real 
distance  from  the  centre  of  the  bolt  is  seen,  and  2"  must  be  the  point  in 
which  the  edge  is  cut  by  the  chamfer.  The  highest  points  in  the  curves  of 
intersection  must  be  at  the  same  level  upon  all  the  faces ;  the  lowest  points 
must  also  be  at  one  level ;  thus  the  arcs  which  represent  the  chamfer  must 
be  tangent  to  a  horizontal  line  drawn  through  i  and  must  end  in  the  edges 
in  points  in  a  horizontal  line  drawn  through  2". 

The  angle  of  the  chamfer  ranges  usually  from  30°  to  45°,  and  much  or 
little  may  be  cut  from  the  head ;  thus  the  circle  left  upon  the  top  may  be 
tangent  to  the  sides,  or  smaller  than  in  Fig.  135. 

FIG.  136.     Three  views  of  a  square  bolt-head  with  a  spherical  top. 

The  top  is  a  portion  of  a  sphere,  and  its  section  by  any  one  of  the  faces 
of  the  bolt-head  is  an  arc  of  a  circle.  The  highest  point  is  seen  at  IF,  and 
the  lowest  point  is  obtained  as  in  Fig.  135. 

The  radius  of  the  spherical  top  is  usually  about  twice  the  diameter  of 
the  bolt. 

FIG.  137.  Three  views  of  a  squart  bolt-head  when  its  vertical  faces 
are  at  an  angle  of  4$°  with  the  front  plane. 

The  arcs  of  circles  at  the  top  of  each  face  appear  ellipses  but  are  repre- 
sented by  arcs  of  circles.  The  lowest  points  in  the  arcs  are  in  the  edges 
and  are  seen  in  both  views.  To  obtain  the  level  of  the  highest  point  re- 
volve the  centre  of  any  face,  as  3T,  to  j'  and  project  to  the  front  view  as 
explained  in  Fig.  135.  The  highest  points  are  in  the  centre  of  each  face  ; 
their  level  is  thus  given  at  3"'  The  arcs  of  both  views  must  be  contained 
between  horizontal  lines  through  if  and  j-^,  as  in  Fig.  135. 


l8o  MECHANICAL   DRAWING. 

FlG.  138.  Three  views  of  a  square  nut  so  placed  as  to  show  two  faces 
in  the  front  view. 

The  nut  is  shown  upon  a  bolt,  which  should  always  be  represented  as 
projecting  slightly  beyond  the  nut.  The  points  for  the  curves  representing 
the  chamfer  are  found  as  has  been  explained. 

A  square  nut  or  bolt-head  should  be  so  placed  as  to  show  one  face  only, 
as  in  Fig.  135. 

The  thickness  of  the  square  nut  and  bolt-head  is  equal  to  the  diameter 
of  the  bolt.  The  distance  between  the  parallel  sides  varies  ;  it  is  often  i  % 
times  the  diameter  of  the  bolt  plus  %  of  an  inch. 

FIG.  139.     Three  views  of  an  hexagonal  bolt-head. 


PLATE  XXVII. 


Fig.  134. 


Fig.  135. 


Fig.  136. 


Fig.  137. 


Fig.  138. 


'CO 


Fig.  139. 


1 82  MECHANICAL   DRAWING. 


PLATE    XXVIII. 

FIG.  140.  Three  views  of  a  bolt  having  an  hexagonal  head  with  a 
spherical  top. 

The  lowest  points  in  the  curves  of  the  chamfer,  which  are  really  arcs  in 
this  figure,  are  in  the  edges  and  are  determined  in  the  front  view  ;  the  highest 
points  are  in  the  centres  of  the  faces  and  are  determined  in  the  side  view. 
The  arcs  which  are  drawn  to  represent  the  chamfer  must  be  tangent  to 
a  line  drawn  through  2,  and  have  their  lowest  points  in  a  line  drawn 
through  /. 

FIG.  141.  Three  views  of  a  bolt,  with  an  hexagonal  head  and  nut, 
whose  corners  are  chamfered. 

The  circle  on  the  top  of  the  head  given  by  the  chamfer  appears,  in  the 
front  and  in  the  side  view,  a  straight  line  whose  length  is  equal  to  the  diame- 
ter of  the  circle.  Drawing  from  the  extremities  of  these  lines  the  elements 
of  the  cone  (generally  at  30°)  determines,  in  the  front  view,  the  lowest  points 
in  the  curve  by  the  intersections  of  these  lines  with  the  edges ;  and  the 
highest  points  are  determined  in  the  side  view,  where  the  lines  are  seen  inter- 
secting the  centres  of  the  faces.  In  this  figure,  and  also  Fig.  140,  the  arcs 
must  end  in  the  edges  at  points  given  by  a  line  drawn  through  point  /,  and 
must  be  tangent  at  the  centre  of  each  face  to  a  line  drawn  through  point  2. 
The  curves  of  the  chamfer  in  this  figure  are  hyperbolas,  as  they  are  in  Fig. 

135- 

The  thickness  of  the  hexagonal  nut  and  bolt-head  is  equal  to  the  diame- 
ter of  the  bolt ;  the  distance  between  the  parallel  sides  varies.  It  is  not 
necessary  to  represent  the  exact  proportions  ;  therefore  many  draughtsmen 
make  the  conventional  drawings  shown  in  Figs.  139,  140,  and  141,  in  which 
the  long  diagonal  of  the  head,  or  nut,  is  made  twice  the  diameter  of  the  bolt. 
This  is  larger  than  the  standard  sizes,  but  is  the  best  representation  for 
practical  drawings. 

When  only  one  view  of  an  hexagonal  bolt-head  or  nut  is  shown,  it 
should  represent  three  faces  of  the  object. 

The  draughtsman  does  not  need  to  find  the  exact  points  for  the  curves 
of  the  chamfer,  as  he  knows  the  appearance  and  can  give  it  by  eye ;  but  it 
is  well  that  students  should  know  how  to  obtain  the  correct  points. 

FIG.  142.     A  spring  or  a  pipe  of  helical  form. 

Make  the  drawing  of  the  helical  curve,  which  is  the  centre  line  of  the 
spring,  as  explained  in  Fig.  126.  With  different  points  in  the  curve  as  cen- 
tres, draw  circles  whose  diameters  are  equal  to  that  of  the  spring.  The 
outline  of  the  spring  must  be  drawn  tangent  to  these  circles. 


PLATE  XKVIIL 


Fig.  140 


Fig.  141. 


DEFINITIONS. 


Altitude.  The  perpendicular  distance  between  the  bases,  or  between 
the  vertex  and  the  base,  of  a  solid  or  plane  figure. 

Angle.  The  difference  in  direction  of  two  lines  which  meet  or  tend  to 
meet.  The  lines  are  called  the  sides,  and  the  point  of  meeting,  the  -vertex 
of  the  angle. 

An  angle  is  measured  by  means  of  an  arc  of  a  circle  described  from  its 
vertex  as  a  centre  and  included  between  its  sides.  The  centre  of  the  arc  is 
the  vertex  of  the  angle. 

If  the  radius  of  the  circle  moves  through  ^^  of  the  circumference, 
it  produces  an  angle  which  is  taken  as  the  unit  for  measuring  angles, 
and  is  called  a  degree. 

The  degree  is  divided  into  sixty  equal  parts  called  minutes,  and  the 
minutes  into  sixty  equal  parts  called  seconds. 

Degrees,  minutes,  and  seconds  are  denoted  by  symbols.  Thus  5  de- 
grees, 13  minutes,  12  seconds,  is  written  5°  13'  12". 

A  RIGHT  ANGLE  is  one  which  is  formed  by  the  radius 
moving  through  ^  of  the  circumference.  It  is  an  angle  of 
90°.  A  straight  angle  is  formed  when  the  radius  has 
moved  over  \  of  the  circumference.  It  is  an  angle  of  180°. 

ACUTE  ANGLE.     An  angle  less  than  a  right  angle. 


OBTUSE  ANGLE.     An  angle  greater  than  a  right  angle. 


OBLIQUE  ANGLE.     One  which  is  not  a  right  or  a  straight  angle. 
REFLEX  ANGLE.     One  which  is  greater  than  1 80°. 

ADJACENT  ANGLE.     Two   angles   are   adjacent   when 
they  have  the  same  vertex  and  a  common  side. 


MECHANICAL   DRAWING.  185 

DIHEDRAL   ANGLE.      The   opening   between    two   intersecting 
planes. 

SOLID  ANGLE.     One  formed  by  planes  which  meet  at  a  point. 
Apex.     The  summit  or  highest  point  of  an  object. 
Arc.     See  Circle. 

Axis  of  a  Solid.  An  imaginary  straight  line  passing  through  its  centre 
and  about  which  the  different  parts  are  symmetrically  arranged. 

Axis  of  a  Figure.  A  straight  line  passing  through  the  centre  of  a 
figure,  and  dividing  it  into  two  equal  parts. 

Axis  of  Symmetry.  A  straight  line  so  placed  in  a  solid  or  a  plane 
figure  that  every  straight  line  meeting  it  at  right  angles  and  extending  in 
each  direction  to  the  boundary  of  the  solid  or  figure  is  bisected  at  the 
point  of  meeting.  In  many  solids  and  plane  figures  an  axis  of  symmetry 
cannot  be  drawn.  • 

Base.  The  opposite  parallel  polygons  of  prisms.  The  polygon  oppo- 
site the  vertex  of  a  pyramid.  The  plane  surfaces  of  cylinders  and  cones. 
The  opposite  parallel  sides  of  a  parallelogram  or  trapezoid.  The  shortest 
or  longest  side  of  an  isosceles  triangle,  and  any  side  in  any  other  triangle, 
but  usually  the  lowest. 

Bisect.     To  divide  into  two  equal  parts. 
Bisector.     A  line  which  bisects. 

Cinquefoil.     A  figure  composed  of  five  leaf-like  parts. 
\ 

Circle.     A  plane  figure  bounded  by  a  curved  line,  called  a 
circumference,  all  points  of  which  are  equally  distant  from  a     a 
point  within  called  the  centre. 

The  boundary  line  is  called  the  CIRCUMFERENCE. 

DIAMETER.     A  straight  line  drawn  through  the  centre,  and  con- 
necting opposite  points  in  the  circumference,  as  a  b. 

RADIUS.     The  distance  from  its  centre   to  the  circumference, 
as  c  e. 

SEMI-CIRCLE.     Half  a  circle,  formed  by  bisecting  it  with  a  diam- 
eter, as  a  db  a. 

ARC.     Any  part  of  the  circumference,  as  e  b. 


1  86  DEFINITIONS. 

CHORD.     A  straight  line  whose  ends  are  in  the  circumference, 
as/jf. 

SEGMENT.     The  part  of  a  circle  bounded  by  an  arc  and  a  chord, 


SECTOR.  The  part  of  a  circle  bounded  by  two  radii  and  an  arc, 
as  b  e  c  b. 

QUADRANT.  A  sector  bounded  by  two  radii  and  one  fourth  of 
the  circumference,  as  a  c  da. 

TANGENT.  A  straight  line  which  meets  a  circumference,  but 
being  produced  does  not  cut  it,  as  k  d.  The  point  of  meeting  is 
called  the  point  of  contact  or  point  of  tangency. 

Circumscribe.  A  polygon  is  said  to  be  circumscribed  about  a  circle 
when  each  side  of  the  polygon  is  a  tangent  to  the  circle  ;  and  a  circle  is  said 
to  be  circumscribed  about  a  polygon  when  th>e  circumference  of  the  circle 
passes  through  all  the  vertices  of  the  polygon. 

Concave.     Curving  inwardly. 

Cone.  A  solid  bounded  by  a  plane  surface  called  the  base,  which  is  a 
circle,  ellipse,  or  other  curved  figure,  and  by  a  lateral  surface  which  is 
everywhere  curved,  and  tapers  to  a  point  called  the  vertex.  Its  base  names 
the  cone.  Thus  a  circular  cone  is  one  whose  base  is  a  circle. 

A  RIGHT  CIRCULAR  CONE  is  generated  by  an  isosceles  triangle 
which  revolves  about  its  altitude  as  an  axis.  The  equal 
sides  of  the  triangle  in  any  position  are  called  elements  of 
the  surface.  The  length  of  an  element  is  called  the  slant 
height  of  the  cone.  Unless  otherwise  stated  "  cone  " 
means  a  right  circular  cone. 

A  FRUSTUM  OF  A  CONE  is  the  part  included  between  the  base 
and  a  plane  parallel  to  the  base  and  cutting  all  the  elements  of  the 
cone. 

A  TRUNCATED  CONE  is  the  part  included  between  the  base  and 
a  plane  oblique  to  the  base  and  cutting  all  the  elements  of  the 
cone. 

Concentric.     Having  a  common  centre. 

Conic  Section.     A  section  obtained  by  cutting  a  cone  by  a  plane. 

Construction.     The  making  of  any  object. 

Construction  Lines.     The  lines  by  which  the  desired  result  is  obtained. 


MECHANICAL   DRA  WING. 


.87 


Constructive  Drawing.  A  drawing  intended  for  the  workman  who  is 
to  make  the  object. 

Contour.     The  outline  of  the  general  appearance  of  an  object. 
Contour  Element.     An  element  which  is  in  the  contour  of  an  object. 

Convergence.  Lines  extending  toward  a  common  point,  or  planes 
extending  toward  a  common  line. 

Convex.     Rising  or  swelling  into  a  spherical  or  rounded  form. 

Corner.  The  point  of  meeting  of  the  edges  of  a  solid,  or  of  two  sides 
of  a  plane  figure. 

Cross-hatched.  In  mechanical  drawing,  a  half  tinting  placed  upon  parts 
cut  by  a  cutting  plane.  In  free-hand  drawing,  the  use  of  lines  crossing  each 
other  and  producing  light  and  shade  effects. 

Curvature.     Variation  from  straightness. 
Curve.     A  line  of  which  no  part  is  straight. 

REVERSED.     One  whose  curvature  is  first  in  one  direction 
and  then  in  the  opposite  direction. 

SPIRAL.  A  plane  curve  which  winds  about  and 
recedes,  according  to  some  law,  from  its  point  of  begin- 
ning, which  is  called  its  centre. 

Cylinder.  A  solid  bounded  by  a  curved  surface  and  by  two 
opposite  faces  called  bases  ;  the  bases  may  be  ellipses,  circles,  or 
other  curved  figures,  and  name  the  cylinder.  Thus  a  circular 
cylinder  (the  ordinary  form)  is  one  whose  bases  are  circles. 

A  RIGHT  CIRCULAR  CYLINDER  is  generated  by  the  revolution 
of  a  rectangle  about  one  side  as  an  axis.  The  side  about  which 
the  rectangle  revolves  is  called  the  height  of  the  cylinder,  also  its 
axis..  The  side  opposite  the  axis  describes  the  curved  surface  of  the 
cylinder,  and  in  any  of  its  positions  is  called  an  element  of  the  surface. 

Cylindrical.     Having  the  general  form  of  a  cylinder. 
Degree.     The  36oth  part  of  a  circumference  of  a  circle. 
Describe.     To  make  or  draw  a  curved  line. 

Design.  Any  arrangement  or  combination  to  produce  desired  results  in 
industry  or  art. 

Develop.     To  unroll  or  lay  out  upon  one  plane  the  surface  of  an  object. 


1 88  DEFINITIONS. 

Diagonal.  A  straight  line  in  any  polygon  which  connects 
vertices  not  adjacent.  5^ — v 

In  regular  polygons,  diagonals  are  called  long  when  they  /  \    \* 
pass  through  the  centre,    as  c  d,  and   short  when  they  extend    V^— X 
between  parallel  sides,  as  a  b. 

Diameter.  See  Circle.  In  a  regular  polygon  with  an  even 
number  of  sides  a  line  joining  the  centres  of  two  opposite  sides 
is  often  called  a  diameter,  as  e  t. 

Edge.  The  intersection  of  any  two  surfaces.  The  boundary  line. 
Edges  are  straight  or  curved,  and  are  represented  by  lines. 

Elevation.  A  drawing  made  on  a  vertical  plane  by  means  of  projecting 
lines  perpendicular  to  the  plane  from  the  points  of  the  object.  The  terms 
elevation,  vertical  projection,  and  front  view  all.  have  the  same  meaning. 

Ellipse.  A  plane  figure  bounded  by  a  line  such  that  the  sum 
of  the  distances  of  any  point  in  it,  as  c,  from  two  given  points 
e  and  f,  called  foci,  is  equal  to  a  given  line,  as  a  b.  The  point 
midway  between  the  foci  is  called  the  centre. 

The  TRANSVERSE  Axis  of  an  ellipse  is  the  longest  diameter  that 
can  be  drawn  in  it,  as  a  b.  It  is  also  called  the  major  or  long  axis. 

The  CONJUGATE  Axis  is  the  shortest  diameter  which  can  be 
drawn,  as  c  d.  It  is  also  called  the  minor  or  short  axis.  The 
foci,  e  and  /",  are  two  points  in  the  long  diameter  whose  distance 
from  c  or  d  is  equal  to  one-half  a  b. 

Face.  One  of  the  plane  surfaces  of  a  solid.  It  may  be  bounded  by 
straight  or  curved  edges. 

Finishing.  Completing  a  drawing,  whose  lines  have  been  determined, 
by  erasing  unnecessary  lines  and  strengthening  and  accenting  where  this  is 
required. 

Foreshortening.  Apparent  decrease  in  length,  due  to  a  position  oblique 
to  the  visual  rays. 

Free-hand.     Executed  by  the  hand,  without  the  aid  of  instruments. 

Frustum.     See  Cone  and  Pyramid. 

Generated.     Produced  by. 

Geometric.     According  to  geometry. 

Half-tint.  The  shading  produced  by  means  of  parallel  equidistant 
lines. 


MECHANICAL   DRAWING.  189 

Hemisphere.     Half  a  sphere,  obtained  by  bisecting  a  sphere  by 
a  plane. 

Horizontal.     Parallel  to  the  surface  of  smooth  water. 

In  drawings,  a  line  parallel  to  the  tog  and  bottom  of  the  sheet  is  called 
horizontal. 

Inscribe.  A  polygon  is  said  to  be  inscribed  in  a  circle  when  all  its 
vertices  are  in  the  circumference  of  the  circle  ;  and  a  circle  is  said  to  be 
inscribed  in  a  polygon  when  the  circumference  of  the  circle  is  touched  by 
each  side  of  the  polygon. 

Instrumental.     By  the  use  of  instruments. 

Lateral  Surface.     The  surface  of  a  solid  excluding  the  base  or  bases. 

Line.  A  line  has  length  only.  In  a  drawing  its  representation  has 
width  but  is  called  a  line. 

STRAIGHT.     One  which  has  the  same  direction  through- 
out its  entire  length. 

CURVED.     One  no  part  of  which  is  straight. 

BROKEN.      One    composed    of    different   successive 
straight  lines. 

MIXED.     One  composed  of  straight  and  curved  lines. 
CENTRE.     A  line  used  to  indicate  the  centre  of  an  object. 
CONSTRUCTION.     A  working  line  used  to  obtain  required  lines. 

DOTTED.     A  line  composed  of  short  dashes.       

DASH.     A  line  composed  of  long  dashes. 

DOT  and  DASH.     A  line  composed  of  dots 
and  dashes  alternating. 

DIMENSION.     A  line  upon  which  a  dimension  is  placed. 

FULL.     An  unbroken  line,  usually  represent- 
ing a  visible  edge. 

SHADOW.     A  line  about  twice  as  wide  as  the  ordinary  full  line. 
A  straight  line  is  often  called  simply  a  line,  and  a  curved  line,  a 
curve. 

Longitudinal.     In  the  direction  of  the  length  of  an  object. 
Model.     A  form  used  for  study. 


1 90  DEFINITIONS. 

Oblique.     Neither  horizontal  nor  vertical. 
Oblong.     A  rectangle  with  unequal  sides. 


Oval.     A  plane  figure  resembling  the  longitudinal  section  of  an 
egg  ;   or  elliptical  in  shape. 

Overall.     The  entire  length. 
Ovoid.     An  egg-shaped  solid. 

Parallel.     Having  the  same  direction  and  everywhere  equally     ~ 

distant.  I    A\ 

Parallelogram.     See  Quadrilateral. 

Pattern.     That  which  is  used  as  a  guide  or  copy  in  making  anything. 
FLAT.     One  made  of  paper  or  other  thin  material. 

SOLID.     One  which  reproduces  the  form  and  size  of  the  object 
to  be  made. 
Perimeter.     The  boundary  of  a  closed  plane  figure. 

Perpendicular.     At  an  angle  of  90°. 

Perspective.  The  art  of  making  upon  a  plane,  called  the  picture  plane, 
such  a  representation  of  objects  that  the  lines  of  the  drawing  appear  to 
coincide  with  those  of  the  object,  when  the  eye  is  at  one  fixed  point  called 
the  station  point. 

DIAGRAM.  An  exact  perspective  drawing  obtained  scientifically 
by  perspective  methods.  It  is  often  very  false  pictorially  when  not 
seen  from  the  station  point. 

PARALLEL.  Diagram  perspective  which  represents  a  cubical 
form  by  the  use  of  one  vanishing  point,  and  represents  by  its  real 
shape  any  face  parallel  to  the  picture  plane. 

ANGULAR.  Diagram  perspective  in  which  two  sets  of  horizontal 
edges  of  a  cubical  form  are  at  angles  to  the  picture  plane,  and  the 
object  is  thus  represented  by  the  use  of  two  vanishing  points. 

OBLIQUE.  Diagram  perspective  in  which,  none  of  the  edges  of 
a  cubical  form  being  parallel  to  the  picture  plane,  it  is  represented 
by  the  use  of  three  vanishing  points. 


MECHANICAL    DRAWING. 


FREE-HAND  or  MODEL  DRAWING.     A  drawing  which,  without 
confining  the  eye  to  the  station  point,  represents  as  far  as  possible 
the  actual  appearance  of  objects.      It  is  made  free-hand,  and  is  for 
most  purposes  more  satisfactory  than  an  exact  diagram  perspective. 
Plan.     Plan,  horizontal  projection,  and  top  view  have  the  same  meaning, 
and  designate  the  representation  of  an  object  made  on  a  horizontal  plane 
by  means  of  vertical  projecting  lines.     In  architecture  it  means  a  horizon- 
tal section. 

Plane  Figure.     A  part  of  a  plane  surface  bounded  by  lines. 
A  plane  figure  is  called  rectilinear  if  bounded  by  straight  lines,  curvi- 
linear if  bounded  by  curved  lines,  and  mixtilinear  if  bounded  by  both 
straight  and  curved  lines. 

Similar  figures  are  those  that  have  the  same  shape. 
Plinth.     A  cylinder  or  prism,  whose  axis  is  its  least  dimen- 
sion.     It  is  circular,  triangular,  square,  etc.,  according  as  it  has 
circles,  triangles,  squares,  etc.,  for  bases. 

Polygon.     A  plane  figure  bounded  by  straight  lines. 

An  EQUILATERAL  POLYGON  is  one  whose  sides  are  all  equal. 
An  EQUIANGULAR  POLYGON  is  one  whose  angles  are  all  equal. 
A  REGULAR  POLYGON  is  one  which  is  equilateral  and  equiangular. 

PARALLEL  POLYGONS  are  those  whose  sides  are  respectively 
parallel. 


TRIANGLE.     A  polygon  having  three  sides  (i). 
QUADRILATERAL.     A  polygon  having  four  sides  (2). 
PENTAGON.     A  polygon  having  five  sides  (3). 
HEXAGON.     A  polygon  having  six  sides  (4). 
HEPTAGON.     A  polygon  having  seven  sides  (5). 


DEFINITIONS. 

OCTAGON.     A  polygon  having  eight  sides  (6). 
NONAGON.     A  polygon  having  nine  sides  (7). 
DECAGON.     A  polygon  having  ten  sides  (8). 
UNDECAGON.     A  polygon  having  eleven  sides  (9). 
DODECAGON.     A  polygon  having  twelve  sides  (10). 

The  centre  of  a  regular  polygon  is  the  common  intersection  of 
perpendiculars  erected  at  the  middle  points  of  its  sides. 

The  polygons  represented  in  the  figures  are  regular  polygons. 

A  Polyhedron  is  a  solid  bounded  by  planes.      It  is  regular  when  its 
faces  are  regular  equal  polygons. 

There  can  be  but  five  regular  polyhedrons  : 

1.  The  TETRAHEDRON,  or  PYRAMID,  which  has  four  triangular 
faces. 

2.  The  HEXAHEDRON,  or  CUBE,  which  has  six  square 
faces. 


3.  The  OCTAHEDRON,  which  has  eight  triangular  faces. 

4.  The  DODECAHEDRON,  which  has  twelve  pentagonal  faces. 

5.  The  ICOSAHEDRON,  which  has  twenty  triangular  faces. 

The  term  hexahedron  is  applied  only  to  a  regular  polyhedron  : 
the  other  terms  may  be  applied  to  irregular  polyhedrons. 

An  infinite  number  of  irregular  polyhedrons,  also  an  infinite 
number  of  other  solids  bounded  by  plane  or  curved  surfaces,  may  be 
conceived. 

Prism.  A  solid  bounded  by  two  equal  parallel  polygons,  having  their 
equal  sides  parallel,  and  by  three  or  more  parallelograms. 

The  polygons  are  called  the  bases  of  the  prism,  the  parallelograms  the 
lateral  'faces,  the  intersections  of  the  lateral  faces,  the  lateral  edges. 

Prisms  are  called  triangular,  square,  pentagonal,  etc.,  according 
as  the  bases  are  triangles,  squares,  pentagons,  etc. 


A  RIGHT  PRISM  is  one  in  which  the  edges  connecting  the 
bases  are  perpendicular  to  the  bases. 


D) 


MECHANICAL   DRAWING.  193 

An  OBLIQUE  PRISM  is  one  in  which  the  edges  connecting 
the  bases  are  not  perpendicular  to  the  bases. 

A  REGULAR  PRISM  is  a  right  prism  whose  bases  are  regular 
polygons. 

A  TRUNCATED  PRISM  is  the  part  of  a  prism  included 
between  the  base  and  a  section  made  by  a  plane  inclined  to 
the  base,  and  cutting  all  the  lateral  edges. 

The  ALTITUDE  of  a  prism  is  the  perpendicular  distance  between 
the  bases. 

The  Axis  of  a  regular  prism  is  a  straight  line  connecting  the 
centres  of  its  bases. 

A  RIGHT  SECTION  of  a  prism  is  a  section  made  by  a  plane 
perpendicular  to  its  lateral  edges. 

A  PARALLELOPIPED  is  a  prism  whose  bases  are  parallelograms 
Produce.     To  continue  or  extend. 
Profile.     The  contour  outline  of  an  object. 

Projection.  Orthographic.  The  view  or  representation  of  an  object 
obtained  upon  a  plane  by  projecting  lines  perpendicular  to  the  plane. 

Pyramid.  A  solid  of  which  one  face,  called  the  base,  is  a  polygon,  and 
the  other  faces,  called  lateral  faces,  are  triangles  having  a  common  vertex 
called  the  vertex  of  the  pyramid.  The  intersections  of  the  lateral  faces  are 
called  the  lateral  edges. 

A  pyramid  is  called  triangular,  square,  etc.,  according  as  its 
base  is  a  triangle,  square,  etc. 

A  REGULAR  PYRAMID  is  one  whose  base  is  a  regular  polygon 
and  whose  vertex  is  in  a  perpendicular  erected  at  the  centre  of  the 
base.  Its  other  faces  are  equal  isosceles  triangles.  The  altitude  of 
any  of  these  triangles  is  called  the  slant  height  of  the  pyramid. 

A  FRUSTUM  of  a  pyramid  is  the  part  included  between       -— .. 
the  base  and  a  plane  parallel  to  the  base  and  cutting  all  the    /  /     \ 
lateral  edges. 

A  TRUNCATED  PYRAMID  is  the  part  included  between       /^^ 
the  base  and  a  plane  oblique  to  the  base  and  cutting  all  the     /  I^S 
lateral  edges. 


194  DEFINITIONS. 

The  Axis  of  a  pyramid  is  a  straight  line  connecting  the  vertex 
and  the  centre  of  the  base. 

The  ALTITUDE  of  a  pyramid  is  the  perpendicular  distance  from 
the  vertex  to  the  base. 
Quadrant.     See  Circle. 

Quadrilateral.  A  plane  figure  bounded  by  four  straight  lines.  These 
lines  are  the  sides.  The  angles  formed  by  the  lines  are  the  angles,  and 
the  vertices  of  these  angles  are  the  vertices  of  the  quadrilateral. 

A  PARALLELOGRAM  is  a  quadrilateral  which   has  its  opposite 
sides  parallel. 

A  TRAPEZIUM  is  a  quadrilateral  which  has  no  two  sides 
parallel. 

A  TRAPEZOID  is  a  quadrilateral  which  has  two  sides, 
and  only  two  sides,  parallel. 

A   RECTANGLE  is  a  quadrilateral  whose   angles   are 
right  angles. 

A  SQUARE  is  a  rectangle  whose  sides  are  equal. 

A  RHOMBOID  is   a   parallelogram   whose   angles  are 
oblique  angles. 

A  RHOMBUS  is  a  rhomboid  whose  sides  are  equal. 

The  side  upon  which  a  parallelogram  stands  and  the  opposite  side  are 
called  respectively  its  lower  and  upper  bases. 

Quadiisect.     To  divide  into  four  equal  parts. 

QuatrefoiL     A  figure  composed  of  four  leaf-like  parts.  C*        J 

Section.  A  projection  upon  a  plane  parallel  to  a  cutting  plane  which 
intersects  any  object.  The  section  generally  represents  the  part  behind  the 
cutting  plane,  and  represents  the  cut  surfaces  by  cross-hatching. 

Sectional.     Showing  the  section  made  by  a  plane. 
Sector  and  Segment.     See  Circle. 

Shadow.  Shade  and  shadow  have  about  the  same  meaning,  as  gen- 
erally used ;  but  it  will  be  well  to  designate  by  shadow  those  parts  of  an 


MECHANICAL   DRAWING.  195 

object  which  are  turned  away  from  the  direct  rays  of  light,  while  those  sur- 
faces which  receive  less  direct  rays  and  are  intermediate  in  value  between 
the  light  and  the  shadow  are  called  shade  surfaces. 

CAST.     The  shadow  projected  on  any  body  or  surface  by  some 
other  body. 

Similar  Figures  are  those  which  have  the  same  shape. 

Solid.  A  solid  has  three  dimensions,  length,  breadth,  and  thickness. 
It  may  be  bounded  by  plane  surfaces,  by  curved  surfaces,  or  by  both  plane 
and  curved  surfaces.  As  commonly  understood,  a  solid  is  a  limited  portion 
of  space  filled  with  matter,  but  geometry  does  not  consider  the  matter  and 
deals  simply  with  the  shapes  and  sizes  of  solids. 

Sphere.  A  solid  bounded  by  a  curved  surface  every  point 
of  which  is  equally  distant  from  a  point  within  called  the  centre. 

A  sphere  may  be  generated  by  the  revolution  of  a  circle  about  a  diameter 
as  an  axis. 

Spheroid  (Ellipsoid).  A  solid  generated  by  the  revolution  of 
an  ellipse  about  either  diameter.  When  revolved  about  the 
long  diameter,  the  spheroid  is  called  prolate,  or  the  long 
spheroid  ;  when  about  the  short  diameter,  it  is  called  oblate, 
or  the  flat  spheroid.  The  earth  is  an  oblate  spheroid. 

Spiral.     See  Curve. 

Surface.  The  boundary  of  a  solid.  It  has  but  two  dimensions,  length 
and  breadth. 

Surfaces  are  plane  or  curved. 

A  PLANE  SURFACE  is  one  upon  which  a  straight  line  can  be 
drawn  in  any  direction. 

A  CURVED  SURFACE  is  one  no  part  of  which  is  plane. 

The  surface  of  the  sphere  is  curved  in  every  direction,  while  the  curved 
surfaces  of  the  cylinder  and  cone  are  straight  in  one  direction. 

The  surface  of  a  solid  is  no  part  of  the  solid,  but  is  simply  the  boundary 
of  the  solid.  It  has  two  dimensions  only,  and  any  number  of  surfaces  put 
together  will  give  no  thickness. 

Symmetry.  Design.  A  proper  adjustment  or  adaptation  of  parts  to 
one  another  and  to  the  whole. 

BILATERAL.     Having  two  parts  in  exact  reverse  of  each  other. 


1 96  DEFINITIONS. 

Symmetry.  Geometry.  If  a  solid  can  be  divided  by  a  plane  into  two 
parts  such  that  every  straight  line,  perpendicular  to  the  plane  and  extending 
from  the  plane  in  each  direction  to  the  surface  of  the  solid,  is  bisected  by 
the  plane,  the  solid  is  called  a  symmetrical  solid,  and  the  plane  is  called  a 
plane  of  symmetry.  If  two  planes  of  symmetry  can  be  drawn  in  a  solid, 
their  intersection  is  called  an  axis  of  symmetry.  See  Axis  of  Symmetry. 

The  line  about  which  a  plane  figure  revolves  when  it  generates  a  solid 
of  revolution  is  an  axis  of  symmetry  for  the  solid  ;  it  is  also  called  the  axis 
of  revolution. 

Tangent.  A  straight  line  and  a  curved  line,  or  two  curved  lines,  are 
tangent  when  they  have  one  point  common  and  cannot  intersect ;  lines  or 
surfaces  are  tangent  to  curved  surfaces  when  they  have  one  point  or  one 
line  common  and  cannot  intersect. 

Trefoil     A  figure  composed  of  three  leaf-like  parts. 

Triangle.  A  plane  figure  bounded  by  three  straight  lines.  These  lines 
are  called  the  sides.  The  angles  that  they  form  are  called  the  angles  of 
the  triangle,  and  the  vertices  of  these  angles,  the  -vertices  of  the  triangle. 

Triangles  are  named  by  their  sides  and  angles. 

A  SCALENE  TRIANGLE  is  one  in  which  no  two  sides  are  equal. 

1        An  ISOSCELES  TRIANGLE  is  one  in  which  two  sides  are 
equal. 

An  EQUILATERAL  TRIANGLE  is  one  in  which  the  three 
sides  are  equal. 

A  RIGHT  TRIANGLE  is  one  in  which  one  of  the  angles 
is  a  right  angle. 

An  OBTUSE  TRIANGLE  is  one  in  which  one  of  the 
angles  is  obtuse. 

An  ACUTE  TRIANGLE  is  one  in  which  all  the  angles 
are  acute. 

The  HYPOTENUSE  is  the  side  of  a  right  triangle  opposite  the 
right  angle.     The  other  sides  are  called  the  legs. 


MECHANICAL   DRAWING.  197 

An  EQUIANGULAR  TRIANGLE  is  one  in  which  the  three  angles 
are  equal.  The  value  of  each  angle  is  60°. 

The  BASE  is  the  side  on  which  the  triangle  is  supposed  to  stand. 
In  an  isosceles  triangle,  the  equal  sides  are  called  the  legs,  the  other 
side  the  base;  in  other  triangles  any  one  of  the  sides  may  be  called 
the  base. 

The  ALTITUDE  is  the  perpendicular  distance  from  the  vertex  to 
the  base.     Except  in  the  isosceles  triangle,  there  are  three  altitudes. 
The  vertex  of  the  angle  opposite  the  base  is  often  called  the 
vertex  of  the  triangle. 
Trisect.     To  divide  into  three  equal  parts. 

Truncated.     A  truncated  solid  is  the  part  of  a  solid  included  between 
the  base  and  a  plane  cutting  the  solid  oblique  to  the  base. 
Type  Form.     A  perfect  geometrical  plane  figure  or  solid. 

Vertical.     Upright  or  perpendicular  to  a  horizontal  plane  or  line. 

Vertical  and  perpendicular  are  not  synonymous  terms. 

Vertex.  See  Angle,  Quadrilateral,  Triangle.  The  vertex  of  a  solid  is 
the  point  in  which  its  axis  intersects  the  lateral  surface. 

View.  See  Elevation.  Views  are  called  front,  top,  right  or  left  side, 
back,  or  bottom,  according  as  they  are  made  on  the  different  planes  of  pro- 
jection. They  are  also  sometimes  named  according  to  the  part  of  the 
object  shown,  as  edge  view,  end  view,  or  face  view. 

"Working  Drawing.  One  which  gives  all  the  information  necessary  to 
enable  the  workman  to  construct  the  object. 

Working  lanes.     See  Lines. 


UC  SOUTHERN  REGIONAL  LIBRARY  FAC1UT 


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